The PBR theorem argues that if the quantum state is purely epistemic, then different preparation procedures can correspond to overlapping probability distributions over underlying states, which creates ambiguity about which preparation was used based only on observed statistics. In contrast, if the quantum state is ontic, distinct quantum states correspond to non-overlapping distributions, so in principle one can always infer the preparation given sufficient data. The theorem shows that any model with such overlap cannot reproduce all quantum predictions, and therefore concludes that the quantum state must be ontic.

However, this conclusion relies on the assumption of preparation independence, meaning that independently prepared systems have independent underlying states. If this assumption is relaxed and underlying states can depend on the joint preparation context, then overlap need not occur even in models that are otherwise epistemic. See this paper for example: https://arxiv.org/pdf/1811.01107. In this sense, such models may still be called psi-ontic under PBR’s definition, since distinct wavefunctions correspond to distinct underlying states, but the distinction reflects differences in preparation conditions rather than the existence of distinct physical wave-like entities.

Related work has pointed out that PBR’s criterion can classify intuitively epistemic models as ontic when preparation independence is violated, as discussed in this paper: https://arxiv.org/pdf/2109.02676v2. Other results show more explicitly that by dropping this assumption, one can construct models consistent with quantum mechanics in which different quantum states correspond to the same underlying reality, allowing genuine overlap, as in this paper: https://arxiv.org/pdf/1201.6554.

You should check out this lecture: https://pirsa.org/12050021

Sibling comment is great for PBR. For KS, unfortunately most schools have a blind spot around it. This is partially because KS directly implies the legendary Bell's theorem, which is experimentally testable and has allowed folks to prolong their grief over objective reality by bargaining with those experiments rather than doing maths; teaching Bell's theorem is less of a headache while still breaking classical assumptions. It's also partially because KS is not well-understood as a matter of folklore. I would say that maybe nLab's page on KS states the important part, but they still hedge by using lots of formal language. Here's an informal consequence of KS:

Given. We exist in three spatial dimensions.

Given. Photons of the Standard Model have spin 1.

Corollary (QM is indefinite). The spin of a photon isn't definite prior to measurement.

This leads to the infamous Kochen-Conway theorem, better known as the "free will" theorem: if humans have free will then photons have free will. Kochen has been insistent that this family of results is solid and wants his peers to pay attention; he releases a position paper every few years on the topic, e.g. Kochen 2017 or Kochen 2022.