To value bet or not OTR in a 4bet pot

I think this is a check. Both JJ/QQ got there and no other worse hand gets to the river and calls another bet apart from AhQh.

It's close, but I don't think you can get much of a value bet here. Villain is 3betting a bit wide especially considering it could be a blinds steal. I do think you should be making your 4bet a little bigger, like around $2.00.

As played, you are clearing repping AA, KK, QQ, and AK so what is villain calling you down with? A set is very likely to just call you all the way down and raise the river (if you bet). AJ might if he's not a thinking player, but I think it's unrealistic to get much value on the river here. I'd just check.

The strange thing is that the size of villain's 3 bet is not big, tho he will be OOP. I think he will do this with QQ-99, or even smaller pairs. I do not know if AK will just call...
As I always say, you can always take the safe route and check.
Plus you will get valuable info on villain.

Check it behind and see the showdown.
MJ made a good point, QQ-JJ are good possible holdings given the action pre... so it enhances the checking back benefits.

Really gross spot. It kind of depends how 'bad' villain is, in the sense that some bad nit regulars will open shove JJ/QQ on the river here (even though it makes zero sense for their range). If he's balanced in the least, he'll obviously check this river with all sets for the most part. I think it's probably a check on the river- I don't think AJ is calling and neither is a pocket pair (99/TT/etc).