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PLO100 - turn decision in 3-bet pot vs. LAG with two pair + nfd and 150bb stacks

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PLO100 - turn decision in 3-bet pot vs. LAG with two pair + nfd and 150bb stacks

Blinds: $0.50/$1.00 (4 Players) CO: $85.07
BN: $310.48 (Hero)
SB: $105.77
BB: $147.04
Preflop ($1.50) Hero is BN with A 8 5 4
CO calls $1.00, Hero raises to $3.00, SB folds, BB raises to $10.50, CO folds, Hero calls $7.50
Flop ($22.50) 6 9 4
BB checks, Hero bets $16.00, BB calls $16.00
Turn ($54.50) 6 9 4 8
BB checks

Villain plays 36/21 and 3-bets 9.9 overall. He 3-bets a ton vs. my BTN steal: around 20% from SB and 15% from BB. I reckon I have much more of a limping hand than a raising hand, but since a similar situation might occur with me open-raising instead of iso-raising, I'm very interested in post-flop anyway.

I would rate villain as a very good player, who balances his c/f range on these boards with c/c and c/r. Seems to have very proper frequencies for open-raising, c-betting, folding to c-bets, raising vs. c-bet, c/r, donk-betting, and so on. Both in single-raised and 3-bet pots. Also doesn't auto-stab when checked to, etc. Just very balanced overall (and exploitive when appropriate).

On to the flop. Betting when checked to is a no-brainer. On the turn, with SPR a little over 2, I got confused. On the one hand I figured he'd fold most hands that I'm ahead of (such as over pairs with backdoor clubs, unimproved 9 with side cards, 87 with clubs), and stick it in with hands that are better than mine (such as any straight, or top two + a little something extra). I also think he would've c/r KK/QQ with hearts otf already. All this indicates that checking might be best. On the other hand I felt like checking could cause a lot of trouble on the river (besides improving to a boat or a flush), and that my pot equity + fold equity were sufficient to pot/commit, indicating that a bet might be best. Who can enlighten me? Thanks in advance.

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