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Two main concerns here. (1) both of your ranges are just so narrow, and smash this board completely. I really doubt either one of you have a hand in your range that would be consider a bluff, and if you did, this would be a bad spot to bluff given others range. (2) The Q on the board changes things a bit. AK doesn't look so hot knowing QQ setted, and KK is less likely to put money in as he is beat by all of your value hands, and as said I'm not sure bluffing is occurring often here.
I do believe villain is checking his entire range on this flop. At least I would. And I see the problems with badgers checking his entire range back on this flop, as most turn cards won't change anything, and hands like KK, and possibly AK will likely not put too much money in the pot. However, I just really don't see value in betting AK,KK in badgers position on this flop. And I don't really want my betting range looking just like it is (QQ, AA). So I would check back the flop, and bet turn where villain would be more inclined to think I'm "thinly" vbetting KK, or deciding to extract with AK.
As far as calling in villains shoes, I see no value in calling even 1 street with KK, given badgers range. I highly doubt badgers turns any smaller pp he was setmining with here into a bluff, and some of the hands he calls lightly with preflop (if at all) are now ahead of KK. And AK follows close to the same logic.
I would personally probably check back the flop here, then vbet turn/river. If villain is willing to stack off with AK, I can still get it in on two streets. And I also give him the appearance that I'm weaker. I don't think I get much from KK regardless, but if he's calling a street, he would probably be more inclined to call a street if we check back flop. And well we get the money in whenever it's set v set anyways. So basically, I think our main concern is with how villain plays AK, and to base our line around extracting value from that, while still balancing our range.
But obv just my opinion.
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