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KJs is something like on the 8th percentile of hands? It really can't be a terrible 3-bet.
Personally, I'd prob c/c the flop and most likely c/c down.
The 3 bet is certainly not terrible. I'm just advocating a bit more polarized pre-flop range or at least a flop plan that accounts for having a more condensed pre-flop re-raising range and playing the marginal made hands OOP that go along with it.
The top 8% has to do with all-in equity and not necessarily the non-linear hand ranking we're after to create the best re-raising range in this spot. That said, it's clearly not a huge mistake. I do think that going after the concept of hand ranking for this range in that way is a mistake, though.
After that derail, the issue is that our opponent's range vastly improves on this particular turn relative to our own range. We deal with that by value betting only strong hands and check-calling with a frequency that minimizes exploitation, assuming our opponent is capable of bluffing. Against this player and with how rarely he/she sees the turn and the unlikely nature of such players turning hands into bluffs and/or floating this flop to have enough bluffs on the turn, it looks like a very clear check-fold.
Against a very solid player, this would be a check-call since it's a very strong hand for our range, but one that we can no longer justify value betting.