I think we can rule out all fancy stuff like SC and Axs by preflop action, he called 3bet whan short preflop raise was yet to act, I think that at this point his range is like {AA-22, AK}.
If he had a set he would raise it on flop. AA,KK with heart raises flop or bets turn.
AK without a heart folds on flop, 22-66 folds too.
I also think that AA, KK without a heart would bet turn after calling on flop.
So we are left with
K, A
, QQ, 88, 77.
If he bets QQ for value it's quite thin IMO, and he is likely to check 88,77 behind, figuring that he is likely to win against you in the showdown anyway.
Ok, my conclusion is that it's closer to call than to a fold, as he is unlikely to VB QQ this thin, while he is quite likely to bluff with AK.
P.S. As for not raising flop with AK, I think it's ok, beacuse they are quite deep.