Optimal overlap of epistemic states in $\psi$-epistemic hidden-variable models
In plain words
One view holds that the wave function only describes knowledge of a deeper physical state, so two different wave functions could correspond to the same underlying reality. Theorems show such shared reality must shrink rapidly as the number of quantum levels grows, but the exact limit is unknown.
Precise statement
In an ontological model each pure state psi of a d-dimensional system is a probability measure $\mu_{\psi}$ on an ontic space; the classical overlap $\omega_C(\psi,\phi)$ = integral of $\min(\mu_{\psi},\mu_{\phi})$ is compared with the quantum overlap $\omega_Q=1-\sqrt{1-\mid\langle\psi\mid\phi\rangle\mid^2}$. Psi-epistemic models with $\omega_C>0$ for all non-orthogonal pairs exist in every finite d (Lewis, Jennings, Barrett, Rudolph, PRL 2012), the Pusey-Barrett-Rudolph theorem (Nature Physics 2012) excludes them under preparation independence, and Leifer (PRL 2014) proved that $\omega_C/\omega_Q$ decays exponentially in d for certain state families when $4\ \mathrm{divides}\ d$. Determine the optimal achievable ratio as a function of d: a construction matching the upper bound, or a sharper bound.
What would settle it
A $\psi$-epistemic model whose overlap ratio scaling with $d$ matches a proved upper bound.