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I'd throw in JT and AT flushes. People play a pretty varied distribution of defending strategies here, it seems. For example I think 66/44 are totally in a lot of players' ranges enough that if you aren't sure you should give him a combo or two of those. If you believe he wouldn't 3-bet AK every time, then I would put some AK in his range. It seems like a pretty terrible play to back-raise for value with a wider range than KK+ considering how unbalanced most people's button squeeze ranges are. I'm not sure he should have a back-raising range at all tbh.
So yeah on the flop I would give him some AK, AQ, all the suited broadways, 22-77 with some frequency and 88-QQ with a very high frequency. Against that range betting seems correct. Betting is by far the best play vs the tons of combos of AhQ/AQh AK KQ KJ KT QhQ JhJ ThT 9h9. Checking might slightly improve your EV against the dry pocket pairs and particularly against his missed broadways.
I'm not sure checking improves your EV at all vs his nutted hands, because you gain little pot control; in this spr < 3 pot he's completely capable of betting turn and shoving river if you check back. Checking just lets you catch a 4th heart save a little equity sometimes. Probably the preferred line vs a flush would be to bet 13 and bet/fold the turn with intention of checking back the river.
So your options as I see it are betting smallish to usually bet turn and shove river when the board really blanks off, otherwise often bet bet checking, OR you can bet 19-21 and shove turn. I'm inclined to just hand my stack to a flush and go for a 2 street plan to get max value and protection vs pair/heart and KQ. Probably 18ish on the flop and a slight overbet jam on turn.
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