
612 F.3d 500 (2010)
UNITED STATES of America, Plaintiff-Appellee,
v.
Martin T. WILLIAMS, Defendant-Appellant.
No. 09-3521.
United States Court of Appeals, Sixth Circuit.
Argued: June 8, 2010.
Decided and Filed: July 15, 2010.
*502 ARGUED: Kenneth P. Tableman, Kenneth P. Tableman, P.C., Grand Rapids, Michigan, for Appellant. Carol M. Skutnik, Assistant United States Attorney, Cleveland, Ohio, for Appellee. ON BRIEF: Kenneth P. Tableman, Kenneth P. Tableman, P.C., Grand Rapids, Michigan, for Appellant. Carol M. Skutnik, Assistant United States Attorney, Cleveland, Ohio, for Appellee.
Before: DAUGHTREY, GILMAN, and SUTTON, Circuit Judges.

OPINION
RONALD LEE GILMAN, Circuit Judge.
A jury convicted Martin T. Williams on five counts of fraudulently overbilling Medicare, Medicaid, and several private insurance companies as an employee of a psychiatric medical practice. The district court sentenced Williams to 12 months of probation and ordered him to pay restitution in the amount of $822,459.21. Williams appeals his conviction, as well as the amount of restitution that he was ordered to pay. He also brings a claim of ineffective assistance of trial counsel. For the reasons set forth below, we AFFIRM the portion of the district court's judgment relating to Williams's conviction, DISMISS WITHOUT PREJUDICE Williams's ineffective-assistance-of-counsel claim, VACATE the portion of the district court's judgment that concerns restitution, and REMAND the restitution issue for further consideration.

I. BACKGROUND
In January 2002, Dr. Lal P. Rohira, Sharonne A. Szyrej, and Williams were indicted on one count of conspiracy to commit mail fraud, wire fraud, and healthcare fraud; on three counts of wire fraud; and on one count of healthcare fraud. The relevant conduct took place between approximately July 1991 and September 1997. Szyrej and Williams worked as employees of Rohira, a psychiatrist. All three were charged with illegally billing the federal government's Medicare and *503 Medicaid programs, as well as a number of private insurance companies, primarily at the direction of Rohira. Specifically, the indictment charged Rohira, Szyrej, and Williams with "upcoding" (billing for full therapy sessions when only medicine checkups were provided), billing for scheduled patients who missed their appointments, and billing for therapy services provided by Szyrej, who was not licensed to perform such services.
The district court severed the trial of Williams from the joint trial of Rohira and Szyrej. Szyrej subsequently pled guilty to the conspiracy charge and agreed to testify against her codefendants in exchange for the dismissal of the remaining four charges against her. She later testified as a government witness at the separate trials of both Rohira and Williams.
Rohira was tried before a jury in July 2003 and was found guilty on all five counts. He then filed motions for acquittal and for a new trial. These motions remained pending before the district court when Williams went to trial.
Williams was tried before a jury in December 2003. After the close of the evidence, Williams's trial counsel objected to a proposed jury instruction dealing with the concept of deliberate ignorance. His counsel argued that this instruction conflicted with a later instruction explaining the mental state required to show "the defendant's connection to the conspiracy." The district court overruled Williams's objection and included the instruction in its charge to the jury. Williams was found guilty on all counts.
Three days later, the district court sentenced Szyrej to one year of probation. In addition, the court ordered Szyrej to pay restitution in the amount of $600, concluding that the conspiracy victims had not been able to establish the amount of their losses.
A prior panel of this court described the subsequent procedural history of this case as follows:
Williams filed timely post-trial motions for judgment of acquittal and for a new trial with the court. In the latter filing, he alleged that the prosecution had failed to disclose a letter sent by the government to two prosecution witnesses, a letter that Williams contends constituted a "secret agreement" not to prosecute the witnesses in exchange for their testimony against Rohira and Williams, despite the witnesses' own guilt in the fraudulent billing scheme. According to the defendant, such non-disclosure violated the mandates of Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963), and Giglio v. United States, 405 U.S. 150, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972), Supreme Court decisions requiring the prosecution to provide criminal defendants with evidence in the government's possession that could be considered exculpatory or that could serve to impeach the credibility of prosecution witnesses.
Over a year later, on February 4, 2005, the district court addressed not only Williams's motions for judgment of acquittal and for a new trial, but also Rohira's similar motions that had remained unresolved during the pendency of Williams's trial. In ruling upon those motions in separate decisions, however, the district judge focused upon an issue that had not been raised by either Williams or Rohira. In Williams's case, the district court noted:
The prosecution called FBI Special Agent Graupmann to testify about the amount of financial loss caused by the defendant's alleged billing fraud; through a dubious method, he estimated the loss at over $1 million. Under Booker and Blakely, that is a fact that *504 must be admitted by the defendant or expressly found by the jury beyond a reasonable doubt before it may be used to help convict him or to increase his sentence.

United States v. Williams, 355 F.Supp.2d 903, 908 (N.D.Ohio 2005) (footnote omitted), (citing United States v. Booker, 543 U.S. 220, 125 S.Ct. 738, 160 L.Ed.2d 621 (2005), and Blakely v. Washington, 542 U.S. 296, 124 S.Ct. 2531, 159 L.Ed.2d 403 (2004)). Because the defendant's "jury was never expressly charged with finding the amount of loss," the district judge "concluded that Williams is entitled to a new trial under Booker and Blakely." Williams, 355 F.Supp.2d at 909-10. The court "also noted that the prosecution's alleged Brady violation might entitle Williams to a new trial as well," id. at 904, but, having already relied upon Booker and Blakely to grant relief to the defendant, ruled that "Williams's motion for a new trial on the ground of the prosecution's alleged violation of Brady v. Maryland and his motion for judgment of acquittal due to insufficient evidence are denied without prejudice." Id. at 910.
On February 14, 2005, Rohira filed with the district court a timely motion for reconsideration. In that filing, Rohira asked, in the interest of judicial economy, for a ruling by the court on his unresolved request for judgment of acquittal and for a new trial on grounds that he had originally identified in his post-trial motion, including a Brady violation, prosecutorial misconduct, and ineffective assistance of counsel. Eight days later, on February 22, Williams filed his own motion for reconsideration, referencing "the reasons more fully elucidated" by Rohira.
Concluding that Williams's motion for reconsideration was untimely and, thus, was not an impediment to the district court's February 4 ruling becoming final, the United States filed an appeal to this court on March 3, 2005 (docketed here as No. 05-3293). On August 9, 2005, however, the district judge issued a decision addressing Williams's motion for reconsideration and specifying that the prosecution's failure to "turn over a letter which could readily be construed as a promise of nonprosecution in exchange for the testimony of two key government witnesses" did indeed result in a constitutional violation that entitled the defendant to a new trial. The district court also concluded, however, that Williams's motion for judgment of acquittal should be denied "because the jury's guilty verdicts are supported by substantial and competent evidence." The government then filed a second appeal to this court that challenged only the propriety of the August 9 ruling (docketed here as No. 05-4160). The two matters have now been consolidated for appeal.
United States v. Williams, Nos. 05-3293, 05-4160, 2006 WL 3203748, at *1-*2, ___ Fed.Appx. ___, ___ - ___ (6th Cir. Nov.6, 2006) (brackets and ellipses removed).
With regard to the government's first appeal, this court concluded that the district court erred in holding that Blakely and Booker mandated a new trial for Williams. Id. at *4, ___. The rule set forth by the Supreme Court in Blakely and Booker was not implicated because "the district court never imposed sentence upon [Williams]," and the rule does not otherwise apply to "the jury's resolution of the issues presented to it." Id.
Concerning the government's second appeal, this court determined that because the district court lacked jurisdiction to enter the August 9, 2005 order, that order *505 "was a legal nullity." Id. at *7, ___. The panel noted, however, that "because the district court originally denied without prejudice the defendant's motion for a new trial based upon alleged Brady/Giglio violations, Williams remains free to file any subsequent collateral attack upon his convictions that he deems proper." Id. In conclusion, the court reversed the grant of a new trial in Case Number 05-3293, vacated the judgment in Case Number 05-4160, reinstated Williams's convictions, and remanded the matter for sentencing. Id.
When the district court granted Williams's motions for a new trial and for reconsideration, the court also granted Rohira's motions for the same relief on the same grounds. The government appealed the court's orders as to Rohira but, while that appeal was pending, the parties reached a plea agreement. Rohira agreed to plead guilty to all five counts of the indictment in exchange for a stipulation that, "[a]s a result of the conduct of [Rohira] and his co-conspirators, [the conspiracy victims] suffered a loss of $500,000."
While plea negotiations were taking place between the government and Rohira, the same parties settled a qui tam action filed against Rohira for submitting false claims to Medicare. Rohira agreed to pay $400,000 as a result of this settlement, and the qui tam action was dismissed.
At Rohira's sentencing, the government stated that the "negotiated" total amount of the conspiracy victims' losses was $400,000, and that this loss had been fully satisfied by Rohira's qui tam settlement payment. The government thus represented that the total amount of the loss was $100,000 less than that set forth in the plea agreement, and then (inaccurately) stated that the qui tam settlement payment would be divided up among all victims of the conspiracy, when in fact a Federal Claims Act judgment is payable only to the government and the relator. See 31 U.S.C. §§ 3729(a)(1), 3730(d). Rohira was therefore not ordered to pay any further restitution, and he was sentenced to one year of probation.
Several months after the district court sentenced Rohira, this court reversed the district court's decision to grant a new trial to Williams, as detailed above, and remanded the case for sentencing. On remand, a new Assistant United States Attorney (AUSA) took charge of the case. This new AUSA disagreed with Williams as to the appropriate sentence. The AUSA also took a new position concerning the amount of restitution owed, claiming that the victims' losses had not in fact been fully compensated by Rohira's settlement payment as previously represented, and seeking restitution from Williams in the amount of $822,459.21. Williams argued in response that the government's prior representation that the total amount of the conspiracy victims' losses was paid in full through Rohira's $400,000 settlement estopped the government from seeking additional restitution from Williams. He further contended that the government's methodology was flawed in arriving at its new loss calculations.
After hearing the parties' arguments, the court sentenced Williams to one year of probation and issued an order of restitution against him for the full amount sought by the government, without providing any factual findings concerning the amount of the loss or ruling on Williams's estoppel argument. Williams now appeals both his conviction and the order of restitution.

II. ANALYSIS

A. Deliberate-ignorance instruction
On appeal, Williams challenges his conviction on the ground that the district court gave an improper jury instruction *506 that allowed "the jury to convict Williams of conspiracy based on a theory that he was deliberately ignorant about the conspiratorial agreement." This instruction allegedly "contradicted the court's later instruction that mere knowledge or even approval was not enough to show that Williams was part of the conspiracy."
The government responds that Williams is conflating two separate instructions. It contends that the first instruction dealt with "deliberate ignorance and Williams'[s] knowledge requirement concerning the illegal activity." The second instruction "required the jury to find Williams `knowingly and voluntarily' joined the conspiracy." Thus, the government asserts that the instruction was proper and, even if it was improper, the instruction was harmless error.

1. Standard of review
We review challenges to jury instructions under the abuse-of-discretion standard. United States v. Russell, 595 F.3d 633, 642 (6th Cir.2010). "When jury instructions are claimed to be erroneous, we review the instructions as a whole, in order to determine whether they adequately informed the jury of the relevant considerations and provided a basis in law for aiding the jury in reaching its decision." United States v. Frederick, 406 F.3d 754, 761 (6th Cir.2005). "A judgment may be reversed based upon an improper jury instruction only if the instructions, viewed as a whole, were confusing, misleading, or prejudicial." United States v. Kuehne, 547 F.3d 667, 679 (6th Cir.2008) (citation and internal quotation marks omitted). This means that we "will not reverse on the basis of an erroneous jury instruction if the error was harmless." Frederick, 406 F.3d at 761.

2. Merits of the arguments
The contested deliberate-ignorance jury instruction provided at Williams's trial was as follows:
Required mental state. Ordinarily, there is no way that a Defendant's state of mind can be proved directly, because no one can read another person's mind and tell what that person is thinking. But a Defendant's state of mind can be proved indirectly from the surrounding circumstances. This includes things like what Mr. Williams said or what Mr. Williams did, how Mr. Williams acted, and any other facts and circumstances in evidence that show what was in Mr. Williams'[s] mind.
You may also consider the natural and probable results of any acts that Mr. Williams knowingly did, and whether it is reasonable to conclude that Mr. Williams intended those results. This, of course, is all for you to decide.
No one can avoid responsibility for a crime by deliberately ignoring the obvious. If you are convinced that Mr. Williams deliberately ignored a high probability that the alleged co-conspirators were engaged in the fraud charged, then you may find that he knew that they were engaged in that fraud, but to find this you must be convinced beyond a reasonable doubt that Mr. Williams was aware of a high probability that he was engaged in the fraud charged, and that Mr. Williams deliberately closed his eyes to what was obvious.
Carelessness or negligence or foolishness on their part is not the same as knowledge, and is not enough to convict. This, of course, is all for you to decide.
Williams argues that this instruction confused the jury and contradicted the district court's later instruction concerning the conspiracy charge, which was as follows:

*507 If you are convinced that there was a criminal agreement, you must decide whether the Government has proved that Mr. Williams knowingly and voluntarily joined that agreement to convict. The Government must prove that Mr. Williams knew the conspiracy's main purpose and that he voluntarily joined it, intending to help advance or achieve its goals. This does not require proof that Mr. Williams knew everything about the conspiracy, or everyone else involved, or that he was a member of it from the beginning, nor does it require proof that Mr. Williams played a major role in the conspiracy or that his connection to it was substantial. A slight role or connection may be enough.
But proof that Mr. Williams simply knew about a conspiracy or was present at times or associated with members of the group is not enough, even if he approved of what was happening or did not object to it.
Similarly, just because Mr. Williams may have done something that happened to help the conspiracy does not necessarily make him a conspirator. These are all things that you may consider in deciding whether the Government has proved that Mr. Williams joined this conspiracy. Without more, these are not enough.
What the Government must prove is that Mr. Williams knew the conspiracy's main purpose and that he voluntarily joined it, intending to help advance or achieve its goals. That is essential.
A Defendant's knowledge can be proved indirectly by facts and circumstances which lead to a conclusion that he knew the conspiracy's main purpose, but it is up to the Government to convince you that such facts and circumstances existed in this particular case.
Williams first contends that the deliberate-ignorance instruction is flawed because "the district court substituted `their' for `his' in the last paragraph of the instruction, so that the jury was not given the clear direction that Williams'[s] carelessness, negligence, or foolishness was not the same as knowledge." Because Williams did not raise this objection at trial, we will review the alleged misstatement under the plain-error standard. See Russell, 595 F.3d at 643 (utilizing this standard where a defendant challenged a jury instruction on appeal on a ground that was not raised before the district court).
Although the use of the word "their" in this instruction was incorrect grammatically, we conclude that this minor defect did not render the instruction itself confusing. The instruction still "adequately informed the jury of the relevant considerations and provided a basis in law for aiding the jury in reaching its decision." See Frederick, 406 F.3d at 761. In any event, the district court's use of the word "their" does not constitute plain error. See Russell, 595 F.3d at 643 (explaining that, under plain-error review, this court has "discretion to correct the error only if the error seriously affected the fairness, integrity, or public reputation of the judicial proceedings" (citation and internal quotation marks omitted)).
Williams next argues that the deliberate-ignorance jury instruction "told the jury that it could find Williams'[s] required mental statethat he intended to join the conspiracyeven if he deliberately ignored the conspiracy." But the deliberate-ignorance instruction was provided well before the court's instruction about the elements of a criminal conspiracy. The deliberate-ignorance instruction instead explained the degree of knowledge required by Williams concerning the illegality of his actions and the actions of *508 others. This court has previously held that such an instruction is permissible. See United States v. Warshawsky, 20 F.3d 204, 210 (6th Cir.1994) (explaining that "a `deliberate ignorance' instruction is permissible to show a conspirator's knowledge of the unlawful aims of a conspiracy").
In a completely separate portion of the instructions, after the district court had recited the facts in the indictment underlying the conspiracy charge, the court clearly stated that the government had to prove that "Williams knowingly and voluntarily joined" the conspiracy. Williams does not argue that this instruction was incorrect, and he has failed to demonstrate how this instruction contradicts the court's deliberate-ignorance instruction. He has cited no authority for his argument that a deliberate-ignorance instruction is appropriate in cases where a defendant denies knowledge of the illegal activity, but prejudicial where a defendant admits such knowledge.
There is in fact authority to the contrary. See United States v. Mendoza-Medina, 346 F.3d 121, 134 (5th Cir.2003) (holding that "an error in giving the deliberate ignorance instruction is harmless where there is substantial evidence of actual knowledge" (citation and internal quotation marks omitted)). Williams's argument that the deliberate-ignorance instruction somehow confused the jury about the mental state required to join a conspiracy is accordingly without merit. See United States v. Smigiel, Nos. 97-1571, 97-1577, 173 F.3d 857, 1999 WL 196575, at *3 (6th Cir. Mar.29, 1999) (unpublished table decision) ("Since the deliberate ignorance instruction was clearly directed toward defendants' knowledge of the aims of the conspiracy, the district court was not required to give a specific instruction directing the jury that it could not find conspiracy based upon a deliberate ignorance theory."). And, at worst, any error in giving the instruction was harmless. See United States v. Rayborn, 491 F.3d 513, 520 (6th Cir.2007) (explaining that "even when it is unsupported by evidence, a deliberate ignorance instruction that properly states the law is harmless error").

B. Ineffective assistance of counsel
Williams urges us to vacate his conviction for an additional reason: he contends that his trial counsel was ineffective for failing to timely move the district court for reconsideration of its order on his motion for a new trial before it lost jurisdiction due to the pending appeal. The government replies that the record is insufficiently developed for us to review the claim on direct appeal and, in any event, that Williams's ineffective-assistance claim is meritless.
"As a general rule, a defendant may not raise ineffective assistance of counsel claims for the first time on direct appeal, since there has not been an opportunity to develop and include in the record evidence bearing on the merits of the allegations." United States v. Wunder, 919 F.2d 34, 37 (6th Cir.1990). Such claims "are more properly available in a post-conviction proceeding under 28 U.S.C. § 2255, after the parties have had the opportunity to develop an adequate record on the issue from which the reviewing court is capable of arriving at an informed decision." United States v. Rahal, 191 F.3d 642, 645 (6th Cir.1999).
We recognize that this court provides an exception to this general rule and will review an ineffective-assistance-of-counsel claim on direct appeal where "the record is adequately developed to allow the court to properly assess the merits of the issue." United States v. Fortson, 194 F.3d 730, 736 (6th Cir.1999). But Williams has not provided any substantive argument concerning *509 why the record here is sufficiently developed for us to rule on his ineffective-assistance claim on direct review. We therefore conclude that his claim is premature and decline to depart from the general rule that such a claim should be brought as a post-conviction proceeding pursuant to § 2255.

C. Restitution
Turning now to the issue of restitution, we note that Williams offers four arguments in support of his contention that the district court erroneously ordered him to pay $822,459.21. He first argues that the government should be collaterally estopped from claiming that Williams owes any money beyond the $600 paid by Szyrej and the $400,000 paid by Rohira in restitution because the government previously represented to the court that the total amount of the conspiracy victims' losses was $400,000. Second, Williams asserts that the court violated Rule 32 of the Federal Rules of Criminal Procedure by failing to rule on his objections to the victims' loss calculations as set forth in the Presentence Report (PSR). He next argues that his right to the due process of law prohibits the government from taking conflicting positions concerning restitution as between him and his codefendants. Finally, Williams contends that judicial estoppel prohibits the government from taking such conflicting positions.
The government's primary response to Williams's arguments is that "[t]he defendants in this case were not similarly situated at the time of their sentencing." Specifically, it contends that Rohira and Szyrej were differently situated from Williams because they both pled guilty, whereas Williams elected to proceed to trial. Thus, according to the government, it did not take inconsistent positions regarding restitution during the sentencings of the three codefendants. It further argues that the district court did not violate Rule 32 of the Federal Rules of Criminal Procedure by failing to rule on Williams's objections because "Williams did not file any objections to the Presentence Report."

1. Standard of review
"The propriety and the amount of a restitution order are subject to different standards of review." United States v. Johnson, 440 F.3d 832, 849 (6th Cir.2006). We review de novo "[w]hether a restitution order is permitted under the law." United States v. Bogart, 576 F.3d 565, 569 (6th Cir.2009). If we determine that restitution is permissible, then the amount of restitution ordered by the district court is reviewed under the abuse-of-discretion standard. Id.

2. Relevant statutory law
Both parties acknowledge that the Mandatory Victims Restitution Act of 1996 (MVRA), 18 U.S.C. §§ 3663A-3664, applies to the present case. The MVRA mandates the imposition of restitution as part of the sentence of a defendant who has committed certain crimes. Id. § 3663A(a)(1), (c)(1). Section 3664 sets forth the relevant procedures for applying the MVRA. This section requires the district court to "order restitution to each victim in the full amount of each victim's losses as determined by the court and without consideration of the economic circumstances of the defendant." Id. § 3664(f)(1)(A). The government bears the burden of proving the amount of each victim's loss by a preponderance of the evidence. Id. § 3664(e).
To determine the appropriate amount of restitution, the district court must order the Probation Office to issue a report that includes "information sufficient for the court to exercise its discretion in fashioning a restitution order." Id. § 3664(a). *510 This requires the Probation Office to obtain a list of the victims and the amounts subject to restitution from the AUSA, and then to seek confirmation of these amounts from the identified victims. Id. § 3664(d)(1)-(2). The MVRA further requires that,
[i]f the victim's losses are not ascertainable by the date that is 10 days prior to sentencing, the attorney for the Government or the probation officer shall so inform the court, and the court shall set a date for the final determination of the victim's losses, not to exceed 90 days after sentencing. If the victim subsequently discovers further losses, the victim shall have 60 days after discovery of those losses in which to petition the court for an amended restitution order. Such order may be granted only upon a showing of good cause for the failure to include such losses in the initial claim for restitutionary relief.
Id. § 3664(d)(5).
Any amount of restitution ordered must be "reduced by any amount later recovered as compensatory damages for the same loss by the victim" in a subsequent federal or state civil proceeding. Id. § 3664(j)(2). This is because "the restitution statutes do not permit victims to obtain multiple recoveries for the same loss." United States v. McDaniel, 398 F.3d 540, 555 (6th Cir.2005).
When ordering restitution in a case where the district court determines that multiple defendants have contributed to a victim's loss, "the court may make each defendant liable for payment of the full amount of restitution or may apportion liability among the defendants to reflect the level of contribution to the victim's loss and economic circumstances of each defendant." 18 U.S.C. § 3664(h). In cases where the United States is a victim, "the court shall ensure that all other victims receive full restitution before the United States receives any restitution." Id. § 3664(i). And any such "sentence that imposes an order of restitution is a final judgment notwithstanding the fact" that the sentence may later be modified. Id. § 3664(o).

3. Estoppel
In his first and fourth arguments against the imposition of restitution, Williams contends that the equitable doctrines of collateral estoppel and judicial estoppel barred the government from claiming, as it did during Williams's sentencing proceedings, that the total amount of the conspiracy victims' loss was greater than $400,000the amount set forth by the government during Rohira's sentencing proceedings. The government responds that these doctrines are inapplicable because Rohira and Williams "were not similarly situated at the time of their sentencing." We conclude that the government was not estopped from seeking additional restitution from Williams, but for reasons different from those advocated by the government, as explained below.
Estoppel doctrines are equitable in nature and therefore are "invoked by a court at its discretion." New Hampshire v. Maine, 532 U.S. 742, 750, 121 S.Ct. 1808, 149 L.Ed.2d 968 (2001) (citation omitted) (describing judicial estoppel). Our circuit nevertheless reviews the application of these doctrines de novo. Lorillard Tobacco Co. v. Chester, Willcox & Saxbe, LLP, 546 F.3d 752, 757 (6th Cir. 2008) (judicial estoppel); United States v. Vasilakos, 508 F.3d 401, 405 (6th Cir.2007) (collateral estoppel).
The purpose of equitable doctrines is "to avoid injustice in particular cases." Heckler v. Cmty. Health Servs., 467 U.S. 51, 59, 104 S.Ct. 2218, 81 L.Ed.2d 42 (1984). *511 Courts typically utilize a multi-factored test to determine whether a particular equitable doctrine is applicable. See New Hampshire, 532 U.S. at 750-51, 121 S.Ct. 1808 (citation omitted) (describing a three-factor test for judicial estoppel); Hammer v. INS, 195 F.3d 836, 840 (6th Cir.1999) (setting forth a five-factor test for collateral estoppel). Where the doctrine is invoked against the government, additional considerations are warranted because "it is well settled that the Government may not be estopped on the same terms as any other litigant." Heckler, 467 U.S. at 60, 104 S.Ct. 2218.
The Supreme Court has explained that "[w]hen the Government is unable to enforce the law because the conduct of its agents has given rise to an estoppel, the interest of the citizenry as a whole in obedience to the rule of law is undermined." Id. Accordingly, if estoppel ever applies against the government, it is only in the rare case where "the public interest in ensuring that the Government can enforce the law free from estoppel [is] outweighed by the countervailing interest of citizens in some minimum standard of decency, honor, and reliability in their dealings with their Government." Id. at 60-61, 104 S.Ct. 2218; see also id. at 60 n. 12, 104 S.Ct. 2218 (noting that "at least two of our cases seem to rest on the premise that when the Government acts in misleading ways, it may not enforce the law if to do so would harm a private party as a result of governmental deception").
Williams argues that the government should be bound in the present case by its previous representation that the conspiracy victims' losses had been satisfied by Rohira's qui tam settlement payment. This argument has some plausibility because the government made inaccurate statements to the district court during Rohira's sentencing proceedings, and the court accepted and adopted those statements when it sentenced Rohira.
In Rohira's plea agreement, for instance, the government and Rohira stipulated that "[a]s a result of conduct of Defendant and his co-conspirators, [the conspiracy victims] suffered a loss of $500,000" and that, "under ex post facto principles, due to the dates charged in the indictment, the restitution provision applicable in this case is 18 U.S.C. § 3663 as opposed to 18 U.S.C. § 3663A." Both representations are inaccuratethe former as a factual matter and the latter as a legal matter. Here, the government charged Rohira with participating in a conspiracy that continued after the effective date of the MVRA. See 18 U.S.C.A. § 3663A note (Historical and Statutory Notes, Effective and Applicability Provisions, 1996 Acts). The MVRA therefore applied to Rohira's sentencing. See United States v. Elson, 577 F.3d 713, 721 (6th Cir.2009) (explaining that the MVRA applies to crimes that occurred prior to and continue past the effective date of the MVRA, and that this application does not violate the Ex Post Facto Clause); United States v. Yaker, 87 Fed. Appx. 532, 533-34 (6th Cir.2004) (same).
And the statement in Rohira's PSR that the actual and intended loss amount was approximately $1.8 million does little to remedy this misrepresentation because the MVRA applies to only the actual loss amount. See United States v. Simpson, 538 F.3d 459, 465-66 (6th Cir.2008) ("It is true that the MVRA refers only to `actual' loss, and unlike § 2B1.1 of the Guidelines does not include `intended loss.'" (citation omitted)). Moreover, the MVRA applies regardless of any plea agreement and, significantly, it does not include the provision found in the discretionary restitution statute cited in Rohira's plea agreement that "[t]he court may also order restitution in any criminal case to the extent agreed to *512 by the parties in a plea agreement." 18 U.S.C. § 3663(a)(3).
Rohira's PSR listed six healthcare and insurance providers (including Medicare) as victims of the conspiracy, and inaccurately stated that Rohira's $400,000 qui tam settlement satisfied all restitution issues. At Rohira's sentencing hearing, the AUSA gave the following explanation of the government's position on the losses caused by Rohira's healthcare-fraud conspiracy:
He pleaded guilty to a massive fraud scheme that lasted six years and caused the Medicare and Medicaid programs, as well as numerous insurance companies, more than a negotiated amount, not more than the negotiated amount of $400,000. I submit to you that before Booker and after Booker, the dollar loss is not an element of the crime of conspiracy, of wire fraud, or of healthcare fraud. And that the dollar amount is a lossthe dollar amount of loss is for the Court to decide by a preponderance of the evidence at sentencing. The parties have negotiated that $400,000 amount and submitted it to you.
The insurance companies are not here today complaining, but they were in the witness box at trial, and each one of them came and testified to their loss and to their victim status in this case. And they do wait to be paid, and they will have to divide up the $400,000 that has been negotiated. As to the 3553(a) factors, Dr. Rohira is a highly educated man, who ran his own practice for many years. He has been in this country for 30 years. He controlled every aspect of his practice, including the supervision of the other two individuals; one of whom pleaded guilty, one of whom stood trial in this case, and whose case is presently on appeal.
Based on the above representations and the information in Rohira's PSR, the district court determined that "there is no restitution still remaining." The court also noted in its Statement of Reasons, filed with the Probation Office as a part of the court's sentencing determinations, that there was "no further restitution owed to either Medicare or the defrauded insurance companies." But this was not correct, as demonstrated by the government's arguments during Williams's sentencing.
Because of the government's misstatements, as well as the district court's entry of restitution orders against Rohira and Szyrej based on incomplete and inaccurate information, the procedures utilized in this case to impose restitution on Rohira, Szyrej, and Williams did not comport with the requirements of 18 U.S.C. § 3664. All three either pled guilty or were convicted of conspiracy to commit healthcare fraud, and thus were each responsible for the losses of all victims "directly and proximately harmed" by the conspiracy. See id. § 3663A(a)(2). The total loss suffered by the victims of the conspiracy, even if difficult to ascertain, should have been the same with respect to all three of the codefendants.
But the government's assessment of the total loss did change as to each of them, and the district court did not defer issuing restitution orders applicable to Rohira and Szyrej until more complete information could be obtained, as perhaps it should have. Such deferral is permissible under the MVRA, as demonstrated by the Supreme Court's recent decision in Dolan v. United States, ___ U.S. ___, 130 S.Ct. 2533, 2535, 177 L.Ed.2d 108 (2010). See id. (holding that "a sentencing court that misses the 90-day deadline [for imposing restitution under the MVRA] nonetheless retains the power to order restitutionat least where, as here, the sentencing court *513 made clear prior to the deadline's expiration that it would order restitution, leaving open (for more than 90 days) only the amount"); see also United States v. Vandeberg, 201 F.3d 805, 814 (6th Cir. 2000) (holding that the MVRA's 90-day deadline is not jurisdictional and that the Act "permits amendments to restitution orders to reflect changed circumstances").
Rohira and Szyrej are not, however, before us in the present appealonly Williams is. Our inquiry is therefore limited to whether the above-described procedural missteps preclude the district court from ordering Williams to pay additional restitution. This requires us to determine whether applying an equitable doctrine against the government would be consistent with the MVRA's statutory framework.
In this case, the district court was unable to exercise its discretion to determine how the obligation to pay restitution for the conspiracy victims' losses should be apportioned between the three codefendants due to its (and the government's) failure to comply with the procedural requirements of the MVRA. But the restitution ordered in Williams's case is nevertheless a permissible outcome under the MVRA. This is because the Act provides that each defendant may be required to shoulder the entire restitution burden for the loss caused by multiple defendants. See 18 U.S.C. § 3664(h) ("If the court finds that more than 1 defendant has contributed to the loss of a victim, the court may make each defendant liable for payment of the full amount of restitution or may apportion liability among the defendants....").
Thus, by participating in a conspiracy to commit healthcare fraud, Williams bore the risk of becoming financially responsible for the entire amount of the conspiracy victims' losses. The restitution order in his case may well be disproportionate when compared to that of his codefendants, but when compared to the losses sustained by the innocent victims of the conspiracy, Williams has no basis to complain. See Dolan, 130 S.Ct. at 2535 (noting that "the statute's text places primary weight upon, and emphasizes the importance of, imposing restitution upon those convicted of certain federal crimes").
Despite the mistakes made during the sentencings of Rohira and Szyrej, the government still had a mandatory obligation under the MVRA to seek the full remaining amount of the conspiracy victims' losses as restitution in Williams's case. The equitable doctrines that Williams seeks to have imposed against the government to thwart this obligation, in contrast, are discretionary in nature. Because we are faced with choosing between a statutory obligation and an equitable doctrine, we conclude that the mandatory language of the MVRA trumps the equitable policies underlying the discretionary doctrines of collateral estoppel and judicial estoppel. See, e.g., Dantran, Inc. v. U.S. Dep't of Labor, 171 F.3d 58, 66 (1st Cir.1999) (declining to apply estoppel to "prevent[] the sovereign from enforcing valid laws for no better reason than that a government official has performed his enforcement duties negligently"); United States v. Marine Shale Processors, 81 F.3d 1329, 1348 (5th Cir.1996) (refusing to apply estoppel against the government where "a private party seeks to avoid the coverage of a law because of a government agent's misrepresentation that the law does not apply" and noting "the importance of separation of powers principles to claims of estoppel against the government"); FDIC v. Hulsey, 22 F.3d 1472, 1489 (10th Cir.1994) ("Courts generally disfavor the application of the estoppel doctrine against the government and invoke it only when it does *514 not frustrate the purpose of the statutes expressing the will of Congress or unduly undermine the enforcement of the public laws."); cf. McLean v. NLRB, 333 F.2d 84, 88 (6th Cir.1964) (refusing to apply "the doctrine of waiver or estoppel ... to excuse the employer from its statutory duty to bargain in good faith with the union during the certification period"). The district court therefore did not err in refusing to apply these doctrines against the government in the present case.
Williams acknowledges in his reply brief that "[t]he MVRA requires the court to order restitution in the full amount of each victim's losses as determined by the court, not the parties." But this observation undercuts Williams's argument that he should not have been ordered to pay restitution. This is because the application of estoppel against the government would not extinguish the court's independent duty under the MVRA to obtain information from the Probation Office about the victims' losses so that it may order appropriate restitution. See 18 U.S.C. § 3664(a), (f)(1)(A). The district court therefore did not err in ordering Williams to pay restitution, and we conclude that his arguments to the contrary are without merit.

4. Due process
Williams next argues that constitutional considerations warrant the application of estoppel against the government. Caselaw recognizes that estoppel may be so applied where the constitutional rights of a private party would otherwise be infringed. See, e.g., Ashe v. Swenson, 397 U.S. 436, 445-47, 90 S.Ct. 1189, 25 L.Ed.2d 469 (1970) (determining that the Fifth Amendment's guarantee against double jeopardy requires the application of collateral estoppel to preclude the government from relitigating any issue that was necessarily decided by a jury's acquittal in a prior trial). Williams contends that the Due Process Clause precludes the government from seeking additional restitution against him. But Williams did not raise this argument below. We therefore review this belated claim under the plain-error standard. See United States v. Henry, 545 F.3d 367, 376 (6th Cir.2008) (applying the plain-error standard to an argument raised for the first time on appeal). Under this standard, we will not reverse a challenged decision of the district court unless there was (1) an error, (2) that "was obvious or clear," (3) that affected the substantial rights of the defendant, and (4) that "seriously affected the fairness, integrity, or public reputation of the judicial proceedings." Id. at 376-77.
Williams argues that "[t]he use of inconsistent theories to secure convictions against more than one defendant in proceedings for the same crime violates due process." He cites Stumpf v. Mitchell, 367 F.3d 594 (6th Cir.2004), for this propositiona case in which this court vacated a defendant's plea of guilty and sentence of death, reasoning "that the use of inconsistent, irreconcilable theories to convict two defendants for the same crime is a due process violation." Id. at 611, 616. But the Supreme Court later reversed this court's judgment concerning the defendant's guilty plea, vacated the judgment concerning the death sentence, and remanded for further consideration of the defendant's "prosecutorial inconsistency claim" as it related to his "death sentence in particular." Bradshaw v. Stumpf, 545 U.S. 175, 186-88, 125 S.Ct. 2398, 162 L.Ed.2d 143 (2005); see also id. at 190, 125 S.Ct. 2398 (Thomas, J., concurring) ("This Court has never hinted, much less held, that the Due Process Clause prevents a State from prosecuting defendants based on inconsistent theories."). This court has not yet issued a decision on remand, and no other case in this circuit has determined *515 the issue. See Blalock v. Wilson, 320 Fed.Appx. 396, 418 n. 26 (6th Cir.2009) (noting that there is no "`clearly established' Supreme Court or Sixth Circuit precedent showing that such a prosecutorial strategy would violate a defendant's due process rights").
Moreover, the circuits that have considered the issue have not agreed on whether inconsistent prosecutorial theories violate due process. See United States v. Presbitero, 569 F.3d 691, 702 (7th Cir.2009) (citing cases). And where a defendant challenges an area of the law that has not been conclusively decided by this court or the Supreme Court, a "circuit split precludes a finding of plain error." United States v. Williams, 53 F.3d 769, 772 (6th Cir.1995). Williams has therefore failed to demonstrate that the district court plainly erred under the Due Process Clause when it entered the order of restitution against Williams. See United States v. Coker, 514 F.3d 562, 569 (6th Cir.2008) (rejecting a similar challenge to loss calculations made to determine the defendant's U.S. Sentencing Guidelines range and reasoning that "the government's different loss calculations cannot be prosecutorial misconduct let alone misconduct that rises to plain errorbecause the government has no obligation to stipulate to identical loss amounts with co-conspirators").

5. Rule 32
Finally, Williams urges us to "vacate the restitution order and remand the case for further proceedings" due to the fact that the district court allegedly violated Rule 32(i)(3)(B) of the Federal Rules of Criminal Procedure by failing to address his objections to "the way the restitution was calculated in the presentence report" and to the fact that "there was no reliable way to determine the amount of loss to the victim[s]." The government responds by contending that Williams waived this argument by failing to timely file any written objections to the PSR. It further notes that "[t]he only disagreement Williams expressed with his PSR was his reference to his Sentencing Memorandum at the [s]entencing [h]earing." And in any event, the government asserts that the court "made a factual finding about restitution at sentencing by adopting the loss amounts submitted by the victims in the revised presentence report."
Rule 32(f) of the Federal Rules of Criminal Procedure requires a party to state in writing any objections to the PSR within 14 days of receiving the report. But where a party has not followed this procedure, Rule 32(i)(1)(D) permits the district court "for good cause" to "allow a party to make a new objection at any time before sentence is imposed." Once an objection has been made, Rule 32(i)(3)(B) provides that the court "mustfor any disputed portion of the presentence report or other controverted matterrule on the dispute or determine that a ruling is unnecessary either because the matter will not affect sentencing, or because the court will not consider the matter in sentencing." This court "has consistently construed the requirements of Rule 32 in a strict manner." United States v. Vandeberg, 201 F.3d 805, 814 (6th Cir.2000). As we have explained:
This circuit requires "literal compliance" with this provision. This strict requirement helps to ensure that defendants are sentenced on the basis of accurate information and provides a clear record for appellate courts, prison officials, and administrative agencies who may later be involved in the case. If the sentencing court fails to make these factual findings, we must remand for resentencing.
*516 United States v. Tackett, 113 F.3d 603, 613-14 (6th Cir.1997) (citations and footnote omitted).
In the present case, the government's assertion that Williams never objected to the PSR is inaccurate, and its argument that the district court effectively ruled on all disputed matters by adopting the factual statements in the PSR does not convince us that the court satisfied Rule 32, given the multiple PSRs and the court's ambiguous ruling. See United States v. Gapinski, 561 F.3d 467, 473-78 (6th Cir.2009); United States v. Darwich, 337 F.3d 645, 667 (6th Cir.2003). According to the dates listed on the first page of Williams's PSR, the report was first prepared in January 2004 and then revised the following month. These first two drafts of the PSR listed the total amount of the conspiracy victims' losses at $403,475.56 (according to the government's Sentencing Memorandum). Williams did not file an objection to the PSR at the time, but instead, in January 2004, filed motions for acquittal and for a new trial. His motion for a new trial was granted over a year later, in February 2005. The district court's order was later overturned by this court on appeal, with the mandate issuing in December 2006.
On remand, the district court held two status conferences to discuss restitution. During the second conference, which was held in October 2008, the court expressed its concern about the disparity between the government's recommendations for Williams's sentencing (prison time) and restitution on the one hand and the government's recommendations for Rohira and Szyrej on the other. It then ordered briefing from the two parties on the propriety of imposing a harsher sentence and a much larger amount of restitution on Williams than on his two codefendants.
The government conceded in its Sentencing Memorandum that it informed Williams for the first time on the date of the October 2008 status hearing that the conspiracy victims' losses were greater than the approximately $400,000 reported in the 2004 PSRs. In response, Williams raised an objection concerning the manner in which the PSR calculated the victims' losses due to upcoding. He argued as follows:
In paragraph 12 of the PS[R], the government states that Dr. Rohira would bill for half hour psychotherapy sessions at a rate of $65.00 when actually he performed services worth $40.00. In paragraph 14, the government then calculates this loss from upcoding as 60 patients a week at $85.00 per hour for a total of $984,000. The actual loss to the victims for the upcoding is the difference between the actual service provided and the incorrect charged amount. Thus, the victims were damaged in an amount of $25.00 per hour for all upcoding. Substituting the amount of $25.00 for the $85.00, the correct value is an amount $289,411.65. It is unclear how this amount was dispersed throughout the victim impact calculations, but it does demonstrate that the amount should be less than stated in the PS[R].
The Probation Office then prepared a second revised PSR on March 13, 2009 one week before Williams's sentencing. This PSR was the first to list the total amount of the conspiracy victims' losses at approximately $823,000. During the March 20, 2009 sentencing hearing, the district court asked if there were any unresolved objections to the second revised PSR. Williams's counsel responded that he intended his Sentencing Memorandum, filed in response to the court's request for briefing on restitution, to function as an objection to the PSR. The district court did not directly address this statement, but noted that it had concerns about the *517 amount of restitution sought by the government.
Later during Williams's sentencing hearing, when the court asked if Williams's counsel had anything further to add, counsel responded: "Your Honor, I merely stand by the arguments advanced in our sentencing memorandum." The court said, "All right," but it did not expressly rule on the objections that had been raised in the Memorandum by Williams. Instead, the court simply adopted the PSR's recommendation concerning the amount of restitution that Williams owed.
In summary, significant litigation took place concerning the validity of Williams's conviction between the time that the first two PSRs were prepared in 2004 and the preparation of the final version in 2009. Given the fact that the district court vacated Williams's conviction in February 2005, his failure to object to the 2004 PSRs is irrelevant, to say nothing of the fact that the restitution amount shown in 2004 is essentially equal to what Rohira paid in the qui tam action against him. This court later reversed the district court's grant of a new trial and, on remand, the district court ordered briefing on the issue of restitution. In his Sentencing Memorandum provided in response to the court's order, Williams raised several objections to the much higher amount of restitution that the government was asking the court to impose. The Probation Office then prepared a second revised PSR that incorporated the higher amount of restitution requested by the government, and Williams's counsel informed the court at the sentencing hearing seven days later that he wished his Sentencing Memorandum to be construed as Williams's objection.
We conclude that, under these circumstances, Williams properly raised an objection to the calculation of the conspiracy victims' losses. And unlike in United States v. Vonner, 516 F.3d 382, 388-89 (6th Cir.2008) (en banc), Williams objected to a factual matter over which there was a genuine dispute between the parties. Cf. id. (concluding that the mitigation arguments that the defendant claimed were not considered by the district court did not present "controverted matters," thus leaving "no `dispute' for the district court to have ruled on" (brackets, citation, and internal quotation marks omitted)). In contrast, the district court here was required by Rule 32(i)(3)(B) to "rule on the dispute or determine that a ruling is unnecessary either because the matter will not affect sentencing, or because the court will not consider the matter in sentencing."
But the record does not show any such ruling, nor does it otherwise indicate that the district court fully understood and considered Williams's argument. Cf. Vonner, 516 F.3d at 388 (holding that "[n]othing in the record, or the context of the hearing, suggests that the court did not listen to, consider and understand every argument [the defendant] made" (brackets, citation, and internal quotation marks omitted)). We emphasize that this is not an ordinary case. The government significantly revised the information in the multiple PSRs that it provided to the district court, and the court issued an ambiguous ruling on the calculation of the restitution amount. Given these irregularities, we will vacate the order of restitution and remand the case to the district court so that it may properly rule on Williams's upcoding argument under Rule 32(i)(3)(B) and, after having done so, issue an appropriate order of restitution. See United States v. Lanesky, 494 F.3d 558, 560-61 (6th Cir.2007) (vacating and remanding for resentencing where "the district court did not rule on the disputes at sentencing, but adopted the PSR in its sentencing order").


*518 III. CONCLUSION
For all of the reasons set forth above, we AFFIRM the portion of the district court's judgment relating to Williams's conviction, DISMISS WITHOUT PREJUDICE Williams's ineffective-assistance-of-counsel claim, VACATE the portion of the district court's judgment that concerns restitution, and REMAND the restitution issue for further consideration.
