Make an exploitable fold?

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Make an exploitable fold?

CO: Jon Doe: $0
BN: h0ttyn4eg: $146.55
SB: pagepuga: $222.24
BB: Hero: $108.92
UTG: padde000: $111.40
LJ: ISLS023: $175.24
HJ: Kukoc69: $100.50
Villian is unknown.
Preflop ($1.50) (6 Players)
Hero was dealt A K
padde000 raises to $2.29, ISLS023 folds, Kukoc69 folds, h0ttyn4eg folds, pagepuga folds, Hero raises to $8.50, padde000 calls $6.21
Standard 3-bet for value. Villian snap calls.
This tells me he's probably got something like AQ/AJ/TT/99/KQ or something around that type of strength.
Flop ($18.50) Q 4 4 (2 Players)
Hero checks, padde000 checks
I plan on check-calling this flop. He's got a decent amount of Qx that won't fold, and a medium pair is going to call at least once, too.
Those PPs will check back the flop, his Ahighs might bet, and his Qx probably bet. A balanced player would check back Qx sometimes, but I think he'd do it with KQ/QJ, rather than AQ.

Turn ($18.50) Q 4 4 A (2 Players)
Hero bets $10, padde000 raises to $25.41, Hero calls $15.41
I thought about check-calling about, but I want to get value out of Ax, Qx, protect against FDs and gutshots, and because this is a card I will bluff at also./'
Don't know why I only bet 10.

When he raises now, I think he has AQ almost every time. There is no other hand with showdown value that raises this board. He might do this with AJ, so he can check back the river.
He could have some draws or random bluff, so calling is the only option.
River ($69.32) Q 4 4 A 8 (2 Players)
Hero checks, padde000 bets $29, Hero calls $29

Call or fold river? We're getting over 2:1, and folding this means we pretty fold almost any runout that we checked the flop with.
He can only have 6 combos of AQ, but I think he has it pretty much every time. The half pot bet really makes it look like he wants to get called. Not sure if he would bet AJ this way, probably not.

It's one of those spots where folding is exploitable, and you know what he's got pretty much every time, but can't fold.


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