Limiting law of the maximum of a log-correlated Gaussian field

Let $X_t(x)$ be the log-correlated Gaussian field considered in the source, and let $M'$ be its derivative-martingale measure. Define

$$Y_t=\sup_{x\in[0,1]^d}X_t(x)-\sqrt{2d}\,t+\frac{3}{2\sqrt{2d}}\ln t.$$

Maximum-law conjecture. There is a constant $c>0$ and a limiting random variable $G_d$ such that $Y_t$ converges in law to $G_d$ as $t\to\infty$, with

$$\mathbb{E}[e^{-qG_d}]=c^{-q}\Gamma\left(1+\frac{q}{\sqrt{2d}}\right)\mathbb{E}\left[(M'([0,1]^d))^{-\frac{q}{\sqrt{2d}}}\right].$$

Equivalently, for every real $u$, the conjectured distribution is

$$\lim_{t\to\infty}\mathbb{P}(Y_t\;\leqslant\; u)=\mathbb{E}\left[\exp\left(-c^{\sqrt{2d}}e^{-\sqrt{2d}u}M'([0,1]^d)\right)\right].$$

The centered maximum is known to be tight, and the source derives this prediction heuristically by analogy with branching random walks, where the corresponding result is cited as rigorous. The two formulations are restatements of one conjecture and are merged.