3for3 said
I looked at this hand on PIO.  The solver does not like Villain flop bet size, it much preferred using 70% of pot.  It occasionally check raises your hand on the flop.

The turn is a mix for you, both as a donk, and a check raise, but mostly just check call as you did.  I think it is taking an aggressive line about 10% of the time.

The river is also a mix, but still mostly just check call.  The hand that PIO turns into a bluff on the river with the most frequency is Ad4, which has blockers all over the place, to diamonds; and full houses.  

These assumptions all center around Villains range for a 30% pot bet, when he had the option to bet 70%.  I am going to rerun, and see if we only give Villain one bet size (which I suspect may be the case) how things change. 

It doesn’t change too much, although you take the more passive line you chose almost all of the time.   I guess this is not surprising, since Villain’s range is stronger when he only had 1 bet size, as opposed to choosing the smaller of 2 bet sizes.

PIO raises the nut flush, and better, and still likes a counterfeited A4 as its favorite bluff raises.  

This last part is something to consider in general.  When I am counterfeited, I often go into c/f mode.  A cool play is to check raise those hands, since you block many full houses…  

70% pot on the flop?! That’s actually pretty neat to consider. I don’t think I’d think to go that big. But it makes more sense.

So 90% of the time PIO suggests that we check/call the turn and 10% of the time we lead/ check/raise? Interesting. I am guessing that favoring the passive approach has more to do with the Ace not being in our hand, which is why it occasionally mixes in a check raise on the flop also.

When you say PIO turns Ad4 into a bluff, how does it bluff that hand? By leading, check/raising or a combo of the two?

Your opponent is not going to expect you to check/raise the second nut hand, which could make it a decent hand to check/raise the river with.

That’s super interesting stuff! I totally get the A4 with the Ad. I also see your point on the larger sizing. I do like the idea of larger sizing on these types of boards when there’s a ton of draw hands out there that probably won’t fold (or shouldn’t fold) to a flop bet. So we build the pot here. I would assume if the diamond, a queen, a king, a jack, an 8 come in on the turn we are shutting down, yeah? Or, which of those cards would be better for barreling? I would imagine the diamond would best if we have the Ace of diamonds. Actually yeah now that I think about it a low diamond was literally perfect. I doubt a bigger diamond would have changed much on the turn for the guy with the Ace of diamonds but it could be a deal breaker for the river. Very interesting stuff.

Foucault said
Couple guesses without looking closely at Pio details:

1. Aggression is low in general on three-flush boards. It’s hard to bluff them without a blocker and when you have a blocker you often have a hand too good to block. It’s also kind of hard to get value with hands like yours because you block Villain’s calling range. Basically blocker effects are just huge relative to other boards.

2. One factor not mentioned w/r/t large c-bet size on flop is two broadway cards. These boards tend to have low betting frequency because stuff like KK/QQ don’t benefit protection, 99 is too far behind to bet, T9 can’t make two overcards fold because they have gutshot. So c-betting range is more polarized than on more dynamic flops where there are lots of ways for bluffs to have equity, thin value bets to deny equity, etc. More polarized range = bigger bet size.  

Andrew, this is super interesting!

1. So you’re saying on the turn here, we should be ranging villain with some sort of blocker hand because like AQ or AK without a diamond wouldn’t (or shouldn’t) be betting because of the board texture? That makes so much sense. If this is true, it’s harder for us to get (more) value from villain because of the fact that he could have us crushed with a better flush, so we benefit from a passive approach. Wouldn’t that also make it harder for an Ace high flush to get value since the only hands that could call are worse kx flushes? I think even a queen high flush would have a hard time calling 3 streets, a J high flush would definitely have a hard time.

2. Basically we can range the cbet in a matter of strictly AK AQ AJ AT with or without the Ace of Diamonds (mainly on the flop). Or KQ KJ JQ with or with out diamonds. Also sets, AA and TTs and 77s I guess. Aside from that range, all other hands would be better played as checks? I think I am understanding that correctly (correct me if I am wrong). So if the more polarized range = bigger bet size, how do we range the smaller bet size? And how do we range the caller on the turn?  Does a T still call the turn bet? I guess T would still be incentivized by the bluffs in villains range, correct?