I would fold the river.

Most villains would bluff the K turn with a hand like AQ AJ. When he checks I think it's possible that he's pot controling a weaker Kx and is going for value on the river.

If he has any other weaker showdown value he would almost always check behind on the river.

Generally bet check bet lines are less likely to be bluffs than when they continue barreling.

I think Turlock summed it up pretty well, some % of the time we may see AQ/AJ on the river, but most of the time it will be just as Turlock said.

Looks like a QQ maybe TT hand. Some AQ yeah maybe I don't think Kx now checks this turn they would be betting to charge draws and underpairs to the K.

I think I would be calling river but not expecting to win more than 35% or so. Having AA AK KQ this would be an odd line for them to check back the turn though it does happen.

Hmm tough one. I'm a little confused as to how people are ranging this? What would you make an anon 3bet v UTG then on this site as that will determine where you go from here. I think I would make it KK+, some SC's, some Axs, maybe AJo or KJs or KQo I suppose.

His flop bet means probably very little as in 3bet pot this flop is not great for calling range as overpairs make a lot of the value range and he has them all vs your capped range. You didn't raise so FD's he can largely discount as well I reckon. I think he would barrel this card with a lot of his air and rep the king. I think he checks here because he has Kx and wants a bet on the river, or he has QQ or JJ and was scared you floated AK.

However when he checks back turn and then you check the river he basically knows you don't have a king. So he can bet his Kx but also his underpairs for thin value (as he has only JJ+ and you might have JJ-99.

AA and KK are also possible but pretty weird. All in all I think I fold this vs an anon.