Critical-temperature variance conjecture for the SK model

Let $F_N^{SK}(\beta)$ denote the free energy of the Sherrington–Kirkpatrick model with system size $N$ at inverse temperature $\beta$, and let the critical value be $\beta_c=1$. Critical-temperature variance conjecture. At the critical value, one expects

$$\operatorname{Var}(F_N^{SK}(\beta_c))=\frac{1}{6}N^{-2}\log N+O(N^{-2}),\qquad\text{as }N\to\infty.$$

This conjecture concerns the fluctuation scale of the SK free energy at the transition between the high- and low-temperature regimes; numerical computations and heuristic arguments support it, while the stated asymptotic remains unproved in the source.