Folding the turn is really tight if the villain is even remotely competent, leading TT+ there to simultaneously fold your equity share and value bet the fish considering how often you're checking behind and how likely the fish is to call with worse would make his range wider than just a full house. Both you and the villain don't reach your points of honesty until the river, the turn is too soon to know for certain KK is no good.
I mean is there a disagreement here about how TT should be played in villain's position, i.e. should villain just check/fold TT assuming hero never double barrels turns as a bluff and the only hands villain "can" lead here are sets? Because frankly there's no way in hell his range is polarized, it's just a question of how thinly he can value bet.