The conjecture on the maximum of the GFF on the circle

Let $V_N$ be the maximum of the discretized Gaussian free field on the circle with singularity parameter $\alpha\geq 0$. Let $M_{(\tau=1,\alpha,\alpha)}$ denote the critical Morris integral distribution, and let $X,Y,Y'$ be the Barnes beta variables defined in the conjecture. Maximum of the GFF on the circle. As $N\to\infty$,

$$\mathbf{E}[e^{qV_N}]\approx e^{q(2\log N-(3/2)\log\log N+\operatorname{const})}\mathbf{E}[M_{(\tau=1,\alpha,\alpha)}^q],$$

and $V_N$ has the stated decomposition into the deterministic centering, $\log X+\log Y+\log Y'+o(1)$. When $\alpha=0$, the $\log X$ term is absent and $Y$ and $Y'$ have the same law. The claim extends the conjecture of Fyodorov and Bouchaud from $\alpha=0$ to general $\alpha$; no resolution is supplied.