Here's my thinking:
Villain is likely to 3-bet preflop with KK+. I think the right process for evaluating this hand is to decide what hands Villain could have here that make sense. So let's start with preflop and narrow down from there.
At 12/8, we know his average range does not include T9s. Specifically, 12/8 looks something like:
Plays: AA-22; AJ+, ATs+, KJ+, KJs+
Raises: 66+; AJ+; AJs+
Even if he opens his range up on the button when there is no raise in front of him, a player this tight is probably calling raises with a range similar to the 8% he is raising with. Possibly include all pairs. Possibly remove AJ/AJs.
I do not believe there are any hands in his range that make the straight. His flush potential comes from AJs, AQs, AKs, and rarely KQs, KJs. He could also have quite a few pocket pairs here. If we think there is a 50% chance that he 3-bets AA/KK, then we can include one and leave the other out. That leaves us with pocket pairs 22-KK (however since there is only one combo of QQ, we can remove it as unlikely enough to be irrelevant).
Our play on the flop is standard, though the bet is a touch on the small side. He is definitely continuing on the flop with 22, 77, 88, 99, TT, JJ, and KK. He is also probably continuing with the suited hands that hit a flush draw. I do not believe a tight player is continuing with a pair lower than 77 on this board against a tight UTG opener.
Thus, on the turn, we beat 99, TT, JJ and we lose to 22, 77, 88, KK, and any of the suited hands that he continued with on the flop. Note that AQs and KQs are out of his range now, since we hold Qd. Perhaps surprisingly, we have 57.5% equity on the turn, in large part due to our 4-flush to the Qd:
Board: 8s 7d 2d 6d
equity win tie pots won pots tied
Hand 0: 42.472% 42.47% 00.00% 598 0.00 { KK, JJ-77, AdKd, AdJd }
Hand 1: 57.528% 57.53% 00.00% 810 0.00 { QdQh }
With that in mind, I think betting the turn is the best play. If we did not have the Qd, it's a bit closer, but we still have 43.5%, so we should probably bet even then. Once we check the turn, we absolutely have to call his bet.
On the river, things are a bit different. But the fact that there are so few hands in his range means that we are still getting good odds to call a bet, even knowing that we only beat JJ, TT, and 99. Even if he only bets JJ in that range, plus the hands that beat us, JJ by itself accounts for enough hands in his range to make a call +EV based on the pot odds:
Board: 8s 7d 2d 6d Kh
equity win tie pots won pots tied
Hand 0: 64.706% 64.71% 00.00% 11 0.00 { KK, JJ, 88-77, AdKd, AdQd, AdJd }
Hand 1: 35.294% 35.29% 00.00% 6 0.00 { QdQs }