I suspect the opposite. The managers will start to panic and then micromanage everything, have crisis meetings etc. This will slow it down even further.
The only real solution to this problem is better long term estimates! The thing is that all estimates are partly BS because we haven't figured out how to predict the future yet. So the solution above is that the supervisor got a BS estimate and they are pushing back with BS progress. There probably is no other way.
Good managers will take the opportunity to learn more. Either there is a way to aid those who made the estimates, a way to help the people doing the work to get the job done, a need to relax the schedule, or some combination of them.
The goal is to get the work done on time and with the resources at hand. Micromanagement is often a hindrance to the goal, so good managers avoid it.
