Is the program required to change the APB to reflect the lower estimate from the ICE?
Based on the background information provided, it would seem that the question is overcome by events since MS C "has been approved." The MDA would have approved of the cost estimate and the DCAPE would have coordinated on it before the ADM was signed. The APB would be updated to reflect the most recent and aggreed upon/coordinated information. Below is some guidance excerpted from the DAG, Chapter 3. I hope this helps.
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220.127.116.11. Cost Assessment Reviews (Pre-Milestone Decisions and Full-Rate Production)
The Office of Cost Assessment (CA), within the Office of Cost Assessment and Program Evaluation (CAPE), receives the results of and reviews all cost estimates and cost analyses and associated studies conducted by the DoD Components for major defense acquisition programs (MDAPs) and major automated information system (MAIS) programs and has timely access to any records and data in the Department.
During the CA review process, CA staff may engage in discussion with the DoD Components regarding any discrepancies related to the cost estimates and comment on deficiencies regarding the methodology or execution of cost estimates. Furthermore, the Director, CAPE, is authorized to concur with the choice of a cost estimate used to support the acquisition program baseline (APB).
18.104.22.168.5. Cost Assessment Review Events—3 Days before Overarching Integrated Product Team (OIPT) Meeting
Cost Assessment (CA) staff's final report is delivered to the OIPT leader at least three days before the OIPT meeting. Immediately thereafter, it is distributed to the OIPT members and is available to the DoD Component staff. The expectation is that any issues had already emerged in prior discussions and that the final CA report should not contain any surprises. The report normally is two to three pages and typically includes the following: •Summary of DoD Component Cost Estimate
•Summary of Cost Assessment independent cost estimate
•Comparison or reconciliation of the two estimates
•Assessment of program risks
•Comparison of (time-phased) Cost Assessment cost estimate to current program funding
•Recommendations concerning program funding