A-K's equity in a typical 4bet pot where villain has just called is poor - ~18% (K-K+). But when giving villain a pretty generous range here (Q-Q+, A-K), we actually about 30% equity.

But, we have basically zero fold equity against hands we are losing to since they called the 4bet and this was an ideal flop for Q-Q+, and have no reason to try to fold out the rest of his likely range (A-K). And we have very poor equity against his calling range to a shove (~17%).

Definitely check/fold the flop. If you hit an Ace/King on the turn, i check/call my stack in.

This may be a good reason to not 4bet A-K deepstacked since this is an ugly spot this deep. With 100BB though against such a range getting it all in on the flop isn't a mistake - as long as you are putting it in first ofc.