Loop-cluster metric conjecture for the continuum loop ensemble

Let $\Sigma$ be a Riemann surface with conformal metric $g$, and let $\mathcal{L}_{g,1/2}$ be the loop ensemble at intensity $1/2$. A cluster is a collection of loops connected by a finite chain of pairwise intersecting consecutive loops. Let $B_1(\Sigma)$ and $B_2(\Sigma)$ denote the two boundary sets used to define the extremal distance $\operatorname{ED}(B_1(\Sigma),B_2(\Sigma))$.

Loop-cluster metric conjecture. There is a metric on the clusters of $\mathcal{L}_{g,1/2}$ inside $\Sigma$, measurable with respect to $\mathcal{L}_{g,1/2}$, obtained as the limit of the pseudo-metrics on metric graphs approximating $\Sigma$. This metric is independent of the time-parametrisation of loops and of the choice of $g$ compatible with the complex structure of $\Sigma$. Moreover, the square of the distance it induces between $B_1(\Sigma)$ and $B_2(\Sigma)$ is exponentially distributed with mean $2\operatorname{ED}(B_1(\Sigma),B_2(\Sigma))$.

The conjecture seeks a conformally natural continuum metric on loop clusters. In a simply connected planar domain, it should agree with the conformally invariant growth metric inside $\mathrm{CLE}_4$, while its existence and the stated distributional description are proposed as scaling limits of the metric-graph formulas.