Several source suggest they don’t.
Berenson, Ginsburg, Christianson, and Yee:
With few exceptions, we learned that Medicare Advantage plans pay provider rates at or close to fee-for-service Medicare rates. Similarly, Medicaid managed care plan payments are close to the relatively low Medicaid fee-for-service rates.
[T]he rates paid for Medicare Advantage enrollees are similar to or slightly above those that Medicare pays for FFS patients’ care.
Medicare Payment Advisory Commission:
MA plan hospital prices are not tied to prices in the non-Medicare market, which is consistent with what we have heard from plans and other market participants. Non-Medicare physician payment rates also appear to have at most a modest relationship to MA bids, suggesting that physician payment rates may be partly anchored to FFS prices.