KQ/KJ check-minraising to see where he is at? Once you checked behind he knows you can't dominate him and pretty much is on weaker K or mid PP? Who knows... If so, I expect him betting the turn (not a shove), yet I can find reasons why he would prefer checking the turn and shoving any river.
For me it looks like too much uncertainty + decisions based on theoretic assumptions on how this unknown plays, really tough marginal spot I would try to avoid
The river is a lottery imo