Nested CLE scaling-limit conjecture for the fuzzy Potts model

Let $q\in (0,4]$ and let

$$\kappa'=4\pi/\arccos(-\sqrt{q}/2) \in [4,8).$$

Suppose that $D$ is a Jordan domain, $\epsilon_n\to 0$ as $n\to\infty$, and that $\mathcal{D}_n$ is a discrete domain in $\epsilon_n\mathbb{Z}^2$ for each $n\geq 1$ such that $\mathcal{D}_n$ converges to $D$ as $n\to\infty$. Let $\omega^n\sim\phi^0_{\mathcal{D}_n,q}$, let $\Gamma$ be a nested $\operatorname{CLE}_{\kappa'}$ in $D$, let $\Gamma^\partial$ be the loops in $\Gamma$ intersecting $\partial D$, and let $\Gamma_{\omega^n}$ and $\Gamma^\partial_{\omega^n}$ denote the corresponding discrete loop collections. Nested CLE scaling-limit conjecture. The pair

$$(\Gamma_{\omega^n}\setminus \Gamma^\partial_{\omega^n},\Gamma^\partial_{\omega^n})$$

converges in distribution to

$$(\Gamma\setminus \Gamma^\partial,\Gamma^\partial)$$

with respect to the metric $d_{\mathcal{L}}$. This conjecture identifies the scaling limit of the nested loop encoding of the fuzzy Potts model; it is known only for $q=2$, corresponding to $\kappa'=16/3$, while the general case remains open.