Interesting TJ hand. Turn play?

[JooWish622]

2-4 NL, 5 handed, villian is fairly tight, althoguh he has reraised from the blinds quite a bit against blind steals.
Hero has a laggy image.

Hero dealt red TJ in SB. VIllian is in BB with 300. I have him covered. Hero raises to 12. Villian calls.

Flop: J94, two hearts. Hero bets 20. Villian raises to 40. Hero calls obviously.

Turn: Black K, Hero bets 40, Villian deliberates, folds.

Hm....

I feel like this stop and go is a conservative route since I would have folded to a raise, but called a bet and POSSBIYLY a river bet and bet a Q river card. I also didn't want to raise since this sometimes A9. But, I feel like he might have folded a worse jack. Given his range of hands, would most of you make this play with TP, mid kicker,against an unknown? Do I want him to fold this turn?

[Big_Jim]

Entirely dependant on what a Min-Raise usually means for this player.

Against many, I would just fold the flop, OOP.

Against players who use it to "find out where they're at" or bluff, I like your line, but I bet closer to the pot.