Scaling-limit conjecture for the metric-graph GFF local-time distance and CLE$_4$

Let $D$ be an open bounded simply connected domain of $\mathbb{C}$. Let $D^\varepsilon$ be a fine-mesh approximation of $D$, $A^{\varepsilon}$ the approximation of the boundary $\partial D$, and $\tilde{\phi}^{\varepsilon}$ the metric graph GFF on $D^\varepsilon$ with zero boundary condition on $A^{\varepsilon}$. Let $\delta_{x,A}(\tilde{\phi}^{\varepsilon})$ denote the local-time distance from $x$ to $A$. Scaling-limit conjecture. As $\varepsilon\to 0$, the local-time distance $\delta_{x,A}(\tilde{\phi}^{\varepsilon})$, jointly with the outermost sign clusters of $\tilde{\phi}^{\varepsilon}$, converges to the time parameters $t(\Gamma)$ on ${\rm CLE}_4$. This conjecture proposes that the second coupling between the Gaussian free field and ${\rm CLE}_4$ arises as the scaling limit of the metric-graph pseudo-metric construction; the source provides no resolution, so the convergence remains open.