[Troll_Inc]

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Given the small sample I am totally not sure how position aware the villain is.

So you think we try to get the money in and if he has KKxx/QQxx then so be it? Check-folding is not an option, right?

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Well if the villain is oblivious to position and is just raising his very best hands, then that increases the chance of KKxx or less likely QQxx. However, if the villain is raising only in LP, then his range greatly increases.

What I'm saying is that this analysis will lead you to the answer of how your hand and that flop play against his range. You'll have to do the math to then determine how you figure to stand in the hand. (I think your guess is right, I can't imagine check folding is the correct play.) But by doing this analysis, you'll have an idea of what to do once you encounter a similar situation with different stack sizes.