Randomly shifted Gumbel limit for the maximum of the two-dimensional inhomogeneous DGFF

Let $d=2$ and suppose that Assumption~-(i) and the moment condition~ hold, in particular $\operatorname{\mathbb{P}}[\omega(e)>0]=1$ for $e\in E_d$. For $n\in\mathbb{N}$, set

$$M_n:=\max_{x\in[-n,n]\cap\mathbb{Z}^d}\varphi_x^{\Lambda_n},\qquad \Lambda_n:=[-n,n]^d,$$

and

$$m_n:=\sqrt{\bar g}\left(\sqrt{2d}\log n-\frac{3}{2\sqrt{2d}}\log\log n\right),$$

where $\bar g$ is as in. Randomly shifted Gumbel limit conjecture. For $\operatorname{\mathbb{P}}$-almost every $\omega$, under $\operatorname{\mathbf{P}}^\omega$, $M_n-m_n$ converges in law to a randomly shifted Gumbel distribution: there exist $\beta^*\in(0,\infty)$ and a random variable $Z$ such that, for every $t\in\mathbb{R}$,

$$\lim_{n\to\infty}\operatorname{\mathbf{P}}^\omega[M_n-m_n\leq t]=\operatorname{\mathbf{E}}^\omega\left[\exp\left(-\beta^*Z\exp(-\sqrt{2d}t)\right)\right].$$

The result would identify the extremal fluctuations of the two-dimensional inhomogeneous DGFF, whose correlations and random conductances make the maximum substantially more intricate than in the homogeneous model. The source presents this as an open problem; it also conjectures that $Z$ is the limit of variables resembling the derivative martingale from critical Gaussian multiplicative chaos and branching processes.