Does a growing static length control relaxation near $T_g$?

Determine whether the point-to-set length $\xi_{\mathrm{PTS}}(T)$ (radius of a cavity, with frozen particles outside it, beyond which the interior forgets the frozen boundary) and the dynamic heterogeneity length $\xi_4(T)$ (from the four-point susceptibility $\chi_4$) are linked to the relaxation time by $\tau_{\alpha} \sim \tau_0 \exp(c \xi^{\psi} / T)$ with a single exponent psi over the range from the onset temperature to $T_g$. An answer gives $\xi_{\mathrm{PTS}}(T)$, $\xi_4(T)$ and $\tau_{\alpha}(T)$ from equilibrium simulations below the experimental $T_g$ and either confirms the relation with a measured $\psi$ or refutes it.

Equilibrium measurements of $\xi_{\mathrm{PTS}}$, $\xi_4$ and $\tau_\alpha$ over enough decades of $\tau_\alpha$ to fix $\psi$ with error bars, or to show that no single relation holds.