Squeeze pot OOP, Turn play after range cbet. Splitting ranges?

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Squeeze pot OOP, Turn play after range cbet. Splitting ranges?

Blinds: $0.05/$0.10 (9 Players) UTG: $10.55
MP: $7.73
UTG+1: $10.15
MP+1: $23.14
MP+2: $11.13
CO: $20.82
BN: $12.22
SB: $10.00 (Hero)
BB: $10.00
Anonymous, no fishy signs so far.
Preflop ($0.15) Hero is SB with 9 9
3 folds, MP+1 raises to $0.20, 2 folds, BN calls $0.20, Hero raises to $1.10, BB folds, MP+1 folds, BN calls $0.90
My squeezing range: 88+,ATs+,KJs+,KQs+, AJo+, A5s.
Calling weaker suited broadways and maybe even baby pairs.
Flop ($2.50) 8 5 2
Hero bets $0.85, BN calls $0.85
I do range cbet here, just following the Nick Howard approach.
Having some fold equity on overcards, ready to 3bet shove against a raise (hmm maybe with tighter ranges it could be an overplay though)
Even if this board is probably better for his range (has all sets and some 8x hands), I doubt that I am sacrificing some huge amounts of ev by range betting.
Turn ($4.20) 8 5 2 3
Hero checks, BN bets $0.70, Hero raises to $3.00, BN calls $2.30
Now it gets interesting, however, I do not want to discuss this particular move, because it just happens once in a blue moon:P
Rather focus on constructing our checking/betting range ( against a normal 1/2-3/4 size)
River ($10.20) 8 5 2 3 Q
Hero checks, BN checks
Final Pot BN wins $9.61
Rake is $0.59

So lets assume our turn range after the range cbet: 88+,ATs+,KJs+,KQs+, AJo+, A5s.
We have 35.3% of overpairs,2.9% of sets, 6.9% of FDs, 7.8% of King high and 51% of A-highs.
I do think its reasonable to check with most AK,AQ hands since they do have a decent showdown value and cant really fold out better, at least OTT. The same applies even for AJ I would say.
Protecting with bigger overpairs which does not need that much of a protection so I would say AA and KK and QQc. Then to correct our value:bluff ratio when betting, I should add the worst combos from vbetting range to my checking range, so 99 and maybe few of TT.
Still not really satisfied with betting ATs, but then I would bluff only KQs and KJs (plus maybe few AcXc). From that point cant I just check my whole range OTT? Or is betting KQs,KJs with some TT-QQ okish?

Personally I do not mind checking range OTT, since I expect pretty much stabs which I can start exploiting by x/raising some overpairs and FDs. However, he might also x/back most of his Ax hands and take a free card vs our overpairs which is my main concern. Soo what is the max ev strategy with the given range OTT?

Any thoughts/advices?:)

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