What is your plan for the river with your range?

Lets say we have JJ-QQ and AQs/AK

I assume that Villain is checking KK/QQ for shure and his bluffs (KQ etc) + Nuts (AA/AK and Ax) some of the time.

Lets say he is checking them around 3/4 of the time. Given that he is not always 3betting AK and QQ lets say 50% of the time he has this range at the river:

Nuts:

AA (4,5 combos)

AK (6 combos)

Ax ( ? )

Bluffcatcher:

KK (6combos)

QQ (0,5 combos)

Bluffs:

KQo (9 combos)

Putting those numbers in crEV a 1/2 Ps bet has the same EV w QQ, when Villain is folding KK and never betting KQo, compared w check/folding them. This is just playing my hand QQ and not my whole range! 

Given that it is pretty close, I like a bet with JJ and a check/fold w QQ. JJ is not blocking KQ and QQ wich gives us a lot more FE and Villain will bluff us more often if we check w KQ.

What do you think, how do you play your Ax hands?