Scaling-limit conjecture for the excursion decomposition of the two-dimensional DGFF

Consider lattice graphs $D_n\subseteq (2^{-n}\mathbb Z)^2$ converging to a bounded, simply connected domain $D\subseteq\mathbb C$, and let $\phi_n$ be the zero-boundary DGFF on $D_n$ converging almost surely to the continuum GFF $\Phi$ in, say, $H^{-\varepsilon}$. Write $(E^n_k,\theta^n_k,\mu^n_k)_{k\geq 1}$ for the excursion decompositions of the DGFFs. Let $\mathds A_{\lambda}$ and $\mathds A_{-\lambda,\lambda}$ denote the continuum level-set objects used to describe the GFF excursion decomposition, with $\lambda$ the relevant boundary value.

Excursion-decomposition scaling-limit conjecture. The discrete decompositions converge to a decomposition $(E_k,\theta_k,\mu_k)_{k\geq 1}$ such that the union of outermost positive clusters is given by $\mathds A_{-\lambda}$ and the union of outermost negative clusters by $\mathds A_{\lambda}$. Individual clusters are obtained by taking $\mathds A_{-\lambda}$ or $\mathds A_{\lambda}$ in the holes of $\mathds A_{-\lambda,\lambda}$ having sign $\lambda$ or $-\lambda$, respectively. Recursively, in holes surrounded by a negative cluster, a positive cluster is given by $\mathds A_{-\lambda}$, while holes surrounded by a positive cluster yield negative clusters by taking $\mathds A_{\lambda}$. Given the clusters, the signs are determined up to a global multiplication by $-1$, and the sign-excursion measures $(\mu_k)_{k\geq 1}$ are the Minkowski-content measures of $(E_k)_{k\geq 1}$ in the same gauge as in Theorem.

This conjecture proposes the continuum scaling limit of the excursion decomposition of the discrete two-dimensional GFF, extending the convergence of the field itself to the joint geometry, signs, and measures of its sign-excursion clusters. The source presents it informally and indicates that a precise formulation would parallel an earlier convergence theorem; no resolution is supplied here.