AJo in 3b pot BB vs BU on A high board - barrel turn for value?

I like it as played. OTT I would x for 6 hits his range more than us and he´s got range advantage now - he´s got 98s, 77, 55 (maybe 66), 76s, A7s, A5s, A6s. We don´t have sets (or at least I don´t). Of strong hands we have AA, A7s, A6s, A5s and that´s about it I guess. I would like to know how much does he stab vs missed T cbet. If he is passive I don´t wanna x and give him a free card, if he´s aggressive I´m definitely down for check.

Btw villain definitely misses value here by not betting OTR.

Thank for the insight. Seems reasonable to me and that´s in line my intuitive approach.

Another question that comes to mind: if villain had bet this river, we would have had to fold? Then what do we call with? We don´t get there with 2pair+... because we bet turn ourselves. So which combos do we call a riverbet with when we take this line? Should we also check turn with AK, so we can catch some bluffs OTR? I think we want to bet AK for value and protection OTT both against Ax and stuff like 87s. So maybe AJ is precisely our bluffcatching hand... Check-call turn and check-call any (?) river vs reasonable sizes?

Well, first ask yourself what are his bluffs OTR? I would say he can have busted hearts and I think 86s would be a good candidate to bluff with there. But then I can´t really see many others. So we really don´t want to have hearts ourselves. It would be very dependant on the sizing he would choose with his R bet (the bigger it is the more hands we can fold). I don´t know the exact answer to your question but I´d say AJ wouldn´t be good enough to call the R bet, I would go with AQ+ (and w/o heart). Also very important is his R aggression. If it were over 28% I would incline to a call much more. But in general ppl are passive on rivers, so we should be slightly overfolding I think.