Aww though spot! If villain is a thinking player he knows ur gonna barrel the K turn alot of the time with your air and therefore it would be a good spot for him to check/raise.
Howeverrrrrrrr after u called the turn check/raise you're range becomes alot stronger ( You represent Kx and draws )
=> Villain can't expect you to fold Kx that often
I would also call the river because we have 2 pair but honestly us having K6 isn't that much of a big difference here compared to a hand like AK/KQ/KJ.
=> The only hand in his range that we beat having K6 and would not beat having AK is 86
86 : 6 combos
88: 3 combos
66: 1 combo
22: 3 combos
A:h x:h : 7 combos ( I assume he 3bets AJs + )
7:h 9:h : 1 combo
79: maybe in his range
You know you can start comparing different combinations to eachother and then start analyzing what's the best on the river.
However you should ask yourself one question which is alot more important than pulling up different combinations etc... :
1) Is villain even capable of bluffshoving the river ? Is he the type ? Not many players are capable of check/raising a K turn and shoving the river with a bluff.
=> Villain might have 500 combinations to bluff and 5 combinations that have you beat
=> But if villain is a nit and never ever bluffing than he has these 5 combinations 100% of the time
To make long answer short:
My bottom calling range is: K2, K6, K8, K4, 86
I fold: Every 1pair hand incl KJ+
( I'll be honest, some of time i'll make a crying call with a hand like KQ but hate life because I know villain isn't bluffing enough in this spot to make that call profitable)