[Sherman]

What to do lies in the answer of this question:

Does BB ever do this with AQ?

If not, I think we have a relatively easy fold. I know I generally advocate never folding AK PF, but that is because it is often impossible to rule AQ out of a villain's range. In this case, it seems that we may have enough evidence (mostly because we are so deep) to rule AQ out of his range.

If so, I think we have a fold. I think most villains would tend to smoother call with hands as strong as AK or JJ here. I think we can be pretty confident given reads, the action, and the stack depth that we are beat here and don't have enough layover to shove it anyway.