[beyonder]

Well about turn check.
Villain check-calls pre, and check-calls flop. Logically he must have a draw, King, Nine, or some PP. He could also have two pairs, set or AK. I mean AK is not likely but not impossible.

Anyway, on turn if i bet:
- Draw will fold and i lose value vs it, or checkraise with a combodraw and then i have to fold the best hand sometimes.
- 9 checkraises probably, maybe even better K.
- Weaker K probably doesnt pay a bet on turn, neither something like TT. And even if they pay turn bet, probably they wount pay river bet, if we bet river after being called twice on this board. So anyway, we probably wount get more than one bet paid, and chances are bigger to get paid on river bet than turn bet.
- Chances of being outdrawn, even by draw, are small and if villain misses a draw, we can extract extra value on possible river bluff from him.

Anyway all considered, I belive turn check in this spot is better than turn bet.