3way action pot: smooth call turn to get UTG to stay in?

I was looking at this HH after my session tonight, and it really stuck with me because it's a dilemma that comes up often: you have the likely best hand against 2 players who both seem more interested in the pot than normal for 3 to the flop action. The trick is to figure out how to get the most value overall, ideally ending in them both going all in and your hand holding up. Of course, that is the ideal and usually doesn't happen, but sometimes it does. But each time you're in this situation, it's up to you to figure out the best approach for maximum value.

In this spot, I had the MP on either AA or KK because he wouldn't minraise the flop w/ QQQ on a rainbow board. He hadn't played a hand in several orbits so I knew that he was dying for action here. I wasn't really sure about the UTG at all, but I doubted that he'd bet out QQQ from UTG on that board, either. And even if he did have it, his relatively short stack ($100) would allow me to at least break even if he did have a higher set and MP continued to go all in.

My real question is, if I smooth call the turn here, will the UTG also call? Of course we don't have the benefit of knowing what he held here, so it's impossible to say for sure. However, I am curious to get some feedback on that scenario, because it does come up and it's obvious of utmost importance to have the best line in the biggest pots. Maybe I just need to hear that I should stop being greedy and be happy w/ the pot, but I just want to make sure that the smooth call or minraise wasn't a better option, given the relative uncoordination of the board (no flushes, and 3-5 or 5-8 are the only hands that really pwn me here). The UTG leadout raise for $10 on the flop tells me at least he's somewhat interested in the hand, and unlikely that it's either of the two achilles heels to my middle set. At the time I was thinking that I didn't want the 4 to the straight to fall and counterfeit my best hand, but now I'm not so sure of that decision.

Ultimate Bet No-Limit Hold'em, $2 BB (6 handed) FTR converter on zerodivide.cx

alias2211 ($856.55)
Button ($174.15)
SB ($221.65)
BB ($294.85)
UTG ($107.70)
MP ($199.90)

Preflop: UTG calls $2, alias2211 is CO with 7, 7.
MP raises to $9, alias2211 calls $9, 3 folds, UTG calls $7.

Flop: ($30) 7, Q, 6 (3 players)
UTG bets $10, MP raises to $20, alias2211 calls $20, UTG calls $10.

Turn: ($90) 4 (3 players)
UTG checks, MP bets $40, alias2211 raises to $210, UTG folds, MP goes all-in for $170.90. alias2211 is
returned $39.10 (uncalled).


Final Pot: $340

Results in white below:
Showdown:

alias2211 shows 7h 7d.
alias2211 has 7h 7d 7c Qh Jh: three sevens.
MP shows Kc Kh.
MP has Kc Kh 7c Qh Jh: a pair of kings.
alias2211 wins $340.

I think a smooth call on the turn is fine, that wasn't exactly a scare card. You have a hand like AQ, AA, KK maybe even QQ all drawing to 2 outs on the river, I don't see a reason to slam UTG out of the pot.

You could consider just calling in hopes of encouraing the other player to join, but he'll fold if he has any idea what's going on. It's fairly obvious you're slow-playing.

The middle position player is already becoming fairly committed, and will be completely so if he bets the river, so raising isn't necessary to win that player's money, but calling may help you win the other player's, so perhaps calling is best, as there's some chance he won't be smart enough to figure out what you're doing.

so smooth call > minraise > larger raise > all in?

OR

smooth call > larger raise > minraise > all in?