Check or Jam this river?

Just jam. It is good that a lot of draws missed because you are more likely to get called. All the overpairs are likely to check back and there are way more of those than busted draws. And even with busted draws, most people underbluff, people are likely checking back with minimal showdown a fair amount of the time. Without a lot of info I would assume he is more like population and not going to bluff river a ton.

Preflop it's close between call if we think we have a big edge on players in the pot and fold. Our hand doesn't do well facing a 4x open 4 ways to the flop.

The pre flop and flop action are a little confusing, but I like the flop check raise. The size seems ok, but I might go a bit bigger, to $116. Just put one $100 chip on top.

Think I'd bet bigger on the turn, like $200-$225.

Given the pre flop and flop action, I might check river to induce, or bet really small, like $40, assuming I understand who did what. It looks like BTN was stabbing flop with a draw that missed, or some low over-pair to the board, like 77/88. He can't call if we jam, but we might induce a bluff if we check or bet super small.

Jam 100%, his range doesn't have all that many full houses (only 99) which would be super sick
you will get called by many overs, if you check he will definitely check overs back

On the other hand, the merit of check calling is that you allow him to bluff, which if he is creative he will over-bluff, but that is rare, and also he calls you with more hands than the hands that bluff you

why check a very polarized range?

I agree with all the comments above, but I would suggest that we have compressed his range considerably on flop and turn. So what is he folding getting 5-2 on the river if we jam?

I think it's a clear jam. I also agree with Docvail that the turn needs to be bigger, given our polar line and the double FD on turn.

Most likely his river range is overpairs and flush draws. I think we're better off checking here and hope he bluffs flush draws - there's just not that many overpairs that will call a river jam here.

Only to clarify with some further thoughts...

The pre-flop action appears to be an EP open to $8, someone in MP calls, BTN calls, and then hero calls in the BB, making the pot $32. Please correct me if that's not right.

On the flop, I like the check to the PFR with the two callers behind. If it was two limpers, then a BTN raise, I think I might prefer a donk bet to a check-raise.

As played, when the PFR checks, and V stabs from the BTN, he could have a pretty wide range here, but it's doubtful he's flatting pre with 99+, rather than 3B'ing. I'd be giving him mostly 77/88 for value, maybe some 1P + a draw combos like 65/54/43, and then just a lot of draws, and un-paired over-cards.

When he stabs for 1/2 pot, and then just calls the check-raise he's unlikely to have 2P or sets, because those would likely 3B for value and protection, unless MAYBE he shows up with 62/42 that's afraid to 3B the flop over an x/r from the BB, because the BB could have all the sets and better 2P.

Those combos of 62/42 are boats on the turn, and have us beat. But I'd think they'd mostly raise turn when we bet large, to avoid a bad runout killing the action.

He's also unlikely to call with his un-paired over-cards with no flush draw. His range is getting pretty compressed now, to 77-88, 65/54/43, and some draws.

Either way, I think the flop x/r could and should be bigger, at least 5x, but we could go 10x at this stack depth, if we want to play a two street game, aiming to jam most turns if the flop x/r gets called.

When hero barrels turn for >2/3 pot, on the 2s, it's unlikely we'd be taking that line with our own over-pairs, and we're not folding out V's over-pairs. It looks like we might be on a flush draw, or have thick value.

V's most likely value hands are going to be 77-88, then some pair + a draw combos that brick out.

The river 9d is mostly a brick, except that now V has to be worried about 9x along with all the other thick value hands in hero's range. If we jam, are 77-88 or any of his 1P + bricked draw combos calling? It seems unlikely, unless we know he's sticky and won't fold any pair here, when all the draws brick.

The only read we have is that V seems decent. If so, he might bluff with his missed draws. It's debatable if he'll bet thin for value if we check or bet small. He's probably not going to hero-call with just 1P if we jam. He probably won't fold his 1P combos if we bet small, and might get curious enough to look us up with ace-high, since we didn't raise pre.

So, to sum up - x/r bigger on the flop, bet bigger on the turn, and then either check or bet stupid-small on the river, to get called (when we bet small), or hopefully induce a bet or raise.

This is not very useful. It's better to know if he is overly passive or overly aggressive - that's important for our river decision.

turn and river bet sizing depends who you're playing against. against someone decent (probably does not apply at 1/2) i dont think you can go too big because IMO your hand is face up as a set or straight once you CR the world here. if i held an overpair like 77 here i would instamuck to you flop CR, but thats just me.