Can this pot be won?

[EnderW27]

You're playing against a passive and predictable opponent.

He limps UTG and you raise 4x in MP with TT. Folded to him and he calls.

Flop comes down Q22 rainbow. Checked to you. You bet 2/3 pot and he smooth calls.

Well he might as well flip his cards over, he's not fooling anyone. He almost certainly has a Q and is 100% beating me right now.

Here's the question: aside from a T coming on the turn or river, are you shutting down completely on this hand? What do you think the liklihood is of pushing him off his Q if a K comes on the turn? How much are you normally willing to invest to see if it's possible?