[Danenania]

His line looks fishy. What Kx hand is he cold calling with PF then check/3-betting flop? Even 55 would normally 3-bet pf I'd think. That leaves 44, Kxc, and perhaps KJo though it's questionable that he'd call KJo pf instead of 3-betting and equally questionable that he'd check/3-bet flop with it. Besides those there are flush draws. I count 13 different unpaired flush draw combos he could very conceivably hold: A8-A6, A3-A2, QJ-Q9, JT-J9, T9-T8, 98.

So on the turn if my earlier assumptions are right then you're just about even money, even without discounting KJo at all. Even if you add in 55 and KT and take out a couple of the flush draw combos you're still getting odds to call down. So I think folding is a pretty bad mistake. You could consider raising/folding to 3-bet but only if he won't bluff missed draws on river, and most opponents will. You can also fold to a river club.