Check it down or bet it out??

[Esoteric]

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No one thinks it's wrong to get Villain off a middle King here? I'm betting somewhere.

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I agree a bet on turn if villain checks twice would be a good line, but not on flop as I think the EV has to be negative since the check behind on the flop gives a chance for the inside straight to hero. Any reason why bet on flop would be good/positive EV?