Optimal overlap of epistemic states in $\psi$-epistemic hidden-variable models

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.

A $\psi$-epistemic model whose overlap ratio scaling with $d$ matches a proved upper bound.