A good rule for OOP 3bet pots is that on turn cards that miss your air range you barrel less often.
4d does nothing for your unmade hands, so you'll check very often. AhKh is not a bad hand to barrel though, still had overcards, blocks QXs, and doesn't block diamonds which is a chunk of your opponents turn folds.

thank you very much

I check turn for the reason already stated. I guess we can bet turn vs villains that overfold turn with all their PP's

Basically a mix I assume. AK has good attributes to justify both options and so we could just hit the RNG and decide.