NL600 Tricky turn spot in squeezed pot

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NL600 Tricky turn spot in squeezed pot

BB: $570.05
UTG: $678.80
HJ: $609
CO: $841.17
BN: $652.80 (Hero)
SB: $600
Preflop ($9.00) (6 Players)
Hero was dealt A K
UTG raises to $18, HJ folds, CO calls $18, Hero raises to $72, SB folds, BB folds, UTG calls $54, CO calls $54
UTG opens 15% folds to 3b 55%
CO calls 10% (3bets vs utg 4,5%)
I consider the CO as good and give him credit for taking creative lines and playing tricky.

I will squeez here pretty tight ,basicаллy AK,QQ+ and occasionaлly some suited connectors or AQo.
Flop ($225.00) 6 9 7 (3 Players)
UTG checks, CO checks, Hero checks
I could cbet here ,but since the range I'd be getting it in against will be fairly strong and I don't think that stackooff with AA here will show great profit,so I choose to checkback as I probably will with my whole range.
Turn ($225.00) K (3 Players)
UTG checks, CO checks
Here is where it gets tricky.I believe that they both will check all of their monsters as I am expected to bet Kh a lot.I think both betting and checking are viable options on this turn and they have their own advantages:

Betting:
+ we can get value from weaker hands (KQs KJs KTs QQ JJ TT 98s 87s). Against those hands we have only one street value I think ,but it might be good to take it here before the board gets too wet (3flush or 4straight).
+we are at the top of our range (but our range is basicly capped at AA,it doesnt beat any of the hands that would raise us of value)

-we could get raised and will be in tough spot as I dont have read on how many bluffs will they have and might be frequently in a situation with 18% equity.the preflopcaller could have all sets and 76s.preflop raiser could still could have some sets.

Checking:
+I'm not too affraid to give them free cards as I am probably in a way ahead/way behind situation and I have implyed odds if I hit flush on the river.I will still get my 1 street of value on most rivers vs KQs KJs KTs QQ JJ TT 98s 87s
+I get to realize my equity against the top of their range (sets/2pairs)

-Might lose value on some bad runouts (4straight or 3fl rivers) vs the weaker part of their ranges,as I will not be able to valuebet (4straight) or they will no longer bluffcatch(if flush hits)
-I miss to build the pot with strong hand(in case someone calls and I river a flush).I dont think this is much of a problem though since I belive the range that is calling me is weak 1pair hands and they are not calling 2 streets on any runout,I will however always get raised by their 2pair/sets.

Thoughts ? Do you prefer betting this turn and with what intention (fold/call vs shove) or checking ?

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