i think we got enough equity to check behind here. too thin to bet/call the turn imo.

Firstly, preflop is fold to me after 3-bet from blind and that would be like quick fold in rush....

Not like I am not loose and like to crack big hands with my speculative ones playing IP HU, but....

I would call preflop with A876ss or K765ss with high FDs, though your T can make non-nut straight with your 6 on 789, but that very "working together T" can bring you a lot of problems if you hit flush or mentioned straight and you face big leading bet(s) from opponent who is OOP.

You flopped pretty decently, but for me its again insta fold cause:

- You draw to non-nutty str8

- You potentially play vs AA+ due to preflop and his atracting/inducing bet size 

- there is possibility you are drawing dead vs A986ds or AA89 or AKK7dd type of hand due to preflop

On turn, his check could mean several things to me:

- c/r preps due to SPR

- bluffcatching c/c with lets say AA or AKQ7 type of hands

- giving up from pot with hand like KKQJ or AKKT with no diamond draw, but its less likely...

- when having AA, he might put you to draw and its often OE such as 68 or 46, so turned 6 could mean to him you still draw when you dont flop wrap with T986 BCS,  but again he could expect you to reraise flop even though there might be chance you play vs AA... Nevertheless he is checking and now he wants to see how much you ll bet OTT...

Finaly, this whole story and decision what to do after his check on turn could have been avoided with easy fold after preflop 3-b of our opponent...

One more thing...

654 is excelent mostly/only  on flops containing 23, but then we have no FE vs AA(5) e.g. And additionally our opponent blocks our 5 outs which could influence on integrity and calculus of our AI shove if we think we play vs lets say KKQJ on flop 238r e.g.

My initial thought was to check back as facing a shove would be awful. But if you are pretty sure you are almost never being shove on then I like a bet for protection/value. 

I think you are getting x-jammed on very infrequently in this spot as I think most villains will almost always bet nuts and 2nd nuts himself on this turn because he doesn't want the turn to check through and this turn card hits you range pretty well. 

I agree that fold pre is saving us a lot of headache, but depending on the villain and how good of a read we have on him. vs a solid player that is barreling with the right frequencies and not afraid to make some light calls I prefer folding pre. 

as played call flop is ok, then I would bet big/cAI OTT because even though we are an underdog vs his strong hands we can also expect a decent amount of c/f , allowing us to pick up a huge pot. in addition we make him fold a lot of hands with equity like A+overs or a bare higher flushdraw etc.

since it's very rare to run into a hand that has us completely crushed (dominating both our 2pairs and our draws) we can't make a big mistake shoving here.
also once we ch back turn we capped our range and would really hate to call any sized bet OTR

think on the flop we're better off raising or folding - if we call there's not a ton of good turn cards and he prob double barrels fairly wide.....Id prob bet the turn - folding out his equity is nice, we get x/r extremely rarely and can make easy river decisions....I feel like checking is close as well but his x/r factor has to be really low