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Against a villain who's willing to overplay the flop with an OESD, it seems to me like he'll frequently overplay the turn as well by leading?

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What percentage of the time does he actually have an OESD, and then what percentage of the time does he fire again on the turn?

Even if he continues on the turn something like 75% of the time, its still a very small chance in the grand scheme of things.

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it still sucks to give up on such a strong hand HU.

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He's telling you pretty clearly that your hand isn't that strong. I really don't like putting anymore money in the pot, but I think c/c c/f is the best way to go about finding out more if you absolutely have to continue.

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If we bet/fold he will tell us all the hands that we are drawing slim against, and call with the hands that we have outs to call with whilst still charging an OESD to draw.

If we check, he will bet KJ, but if we bet he will call KJ. He would also call with hands we beat that he would regularly check through with.

True