Pretty interesting and tough spot. Results?

One question, how the hell is jh a complete brick? If for no other reason it completes a possible flush? But from you saying that I assume that he does not have the flush.

I'd c/f that river, I'm not sure I can identify any hands I beat if he bets.

Sarcasm bro.

X/f river unless villain is likely to turn Qx into a bluff... which is unlikely.

I really don't like a x/f otr... Are we really giving him only sets when he bets? The Jh feels like a much better card to x/c than a black J when we block the nuts (even though it means that he can't be blocking the nuts to bluff). Can't he still have one of the 8 combos of AK, especially the two with Kh, or one of the 6 combos of TT, and a possible 6 combos of 99 (if he raises that MP)? I just feel like if this guy has a tendency to pull off some big bluffs, this is the river he's going to try it on. And I see people occasionally check back sets in a spot like this, not believing that they can get any value from overpairs (and fearing the crai).

None of the hands you mentioned make sense. Sets/flush combos make sense and maybe even QJ.

According to OPs description, villain should have no flushes or QJ. I feel like a float with AK or a flop call and turn bet just to take it down with 99 or TT makes sense (as much sense as JJ at least). If he never has 99 or TT, then he probably never has JJ. So we're giving villain exactly 3 combos of QQ and zero combos of anything else he might bet as a bluff? It doesn't fit description of villain as a "strong player" who shouldn't be flatting hands like TT/JJ or AK and just always giving up to a single flop bet when he misses his set or TPTK...

If he's good he's checking back 99, 1010 and JJ. He has show down value. It's pretty bizarre that you'd remove all flush combos from villains range.

99 and TT don't really have SDV against us on the river... If I bet turn, I'd have no problem c/f this river, but when I start checking aces against a villain known to pull off large bluffs, I'm not really doing so intending to fold. Also, according to OP's description, villain should almost never have a flush here. KThh or 9Thh raise/call from MP when we put him mostly on TT+/AK pre? Bottom line, I actually prefer a heart to check/call here as a lot of villains will be polarized to a flush or a bluff on this river...

Results:

I checked because I also thought betting would be weird. In hindsight though a $500 bet would scream AKhh and would fit with the 3 bet/bet/check line. V bet $500. I tanked for quite a while. Ultimately I concluded that he never has a flush and that sets would be afraid of the hearts and might check. I called and he showed JJ. Obviously I got unlucky but I'm still not sure if I compounded the error with my turn check. As played though I really felt like I needed to call and expected to see KK or 1010 a substantial part of the time.

I doubt villain would bet KK here, but TT and AK both seem possible considering how he got to the river with JJ... I think the check/call here is ok. You're getting like 2.6 to 1 I think, so just need to be good like 28% of the time...

If his range is TT+ here (and TT should get here about as often as JJ), and he never bets KK or AA, then he needs to bluff with TT just a bit over 50% of the time. Maybe that's pushing it, but I think we get to 28% if we start adding in 99 and AK, and he checks back sets sometimes. Need to fold two black aces though.

However, this is the kind of river that I might do weird things like bet $100 - it confuses villain just enough to not raise sets a lot of the time (obviously you can't always make blocking bets like this, or he'll figure it out - and if you do make really small block bets, sometimes you should use this to try to induce raises in spots you might otherwise check/raise). It would also be a pretty sick spot to check-raise all-in as a bluff (only against good players), but stacks would probably need to be a bit deeper.

CRAI might be an option if we were deeper AND it wasn't spread. Otherwise this would be just a raise of $500 into a pot of almost $2400. The odds would be too good to pass up to call. It's rare but there are times the $500 max bets affects the action.