Derivative-chaos law conjecture for the maximum of log-correlated Gaussian fields

Let $(X_n)_n$ be a cutoff approximation of a centered Gaussian field on $[0,1]^d$ with logarithmic covariance, and set $c_n={\rm Var}(X_n)$. Let $M'$ be the derivative chaos and let $G_d$ be a random variable. Derivative-chaos maximum conjecture. As $n\to\widetilde\to ginfty$, one has, in law,

$$\sup_{x in [0,1]^d} X_n(x)-\sqrt{2d}\,c_n+\frac{3}{2\sqrt{2d}}\ln c_n \to G_d,$$

and there is a constant $c>0$ such that

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

The second-order maximum asymptotics are proved for the discrete Gaussian free field in dimension two, while the stated generality is presented as a conjecture by analogy with branching random walks.