I think villain is more likely to raise/fold 1p hands that we beat OTR than call with them. That really depends if we're betting Ax OTR though. What do we do with AT in this spot?
Aren't most regs at this level going to to either raise or fold if hero bets river?
It largely depends how thinly Villain thinks Hero value bets the river and how often he'll bet fold.
I doubt you'd see only shoves or folds though because the ace is a decent barrel card so it will often polarize player's river betting ranges somewhat (they won't have many/any 9x for value) so those borderline hands don't gain much by shoving since they either run into a monster (hand that gladly or begrudgingly calls) or air.