[I dunno]

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I have my doubts that checking can induce a bluff here. You made a very strong move on the turn, and the pot is offering you four to one to call, so Villain would need strong evidence that a bluff would work here before putting it in with air.

If you plan to call any bet here, you're giving the negative freeroll, where villain is more likely to bet only the hands that have you beat, and check the likely losers. Given your turn rep, putting the last bit in has a small chance of knocking out a weak flush that is better than your weak flush. Granted, it's likely V will call with a slightly better flush, but it's still better than checking and watching the 7-high flush check and collect.

(Again, I don't play this rich; grain of salt.)

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People generally don't make it this high without being aggressive. He's probably not going to bluff here that often, but I'm sure every now and then he's shoving some whiffed hand. Being more aggressive means getting away with thinner value bets, and this is certainly a spot where he could with any flush. He's probably correct to bet the nut straight for value as well.

My first post was kind of scrambled, but I think it comes down to this. Villain is going call or bet with any flush. He also probably bets or calls with the nut straight. But if you check, you allow him to occasionally bluff into you, which happens more often than you will get looked up by a good two pair, which he's unlikely to have anyway.