Consistency of Blockchain After Committee Rotation (on HotStuff)

I know that the consensus protocol of Aptos is based on HotStuff (with some modifications). Also, for performance reasons the protocol will be used with an elected committee and not with the entire network.
My question is - how do you guarantee the blockchain consistency after switching the committee? HotStuff can be proven to be consistent (no conflicting decisions) as long as you have the same committee (this is due to the locking mechanism that is used). However, once you switch a committee, theoretically it might be that the vast majority of it are not aware to the last block that was generated, thus they can approve a new block that conflicts with that one. Is there any solution to the problem? Or some hidden assumption used?