The conformal-invariance conjecture for FK percolation

Fix $q \in [1,4)$ and $r \in (0,1)$, and let $D$ be a simply connected domain. Let $D_n=(V_n,E_n)$ be discrete domains in $\frac{1}{n}\mathbb{Z}^2$ converging to $D$. For critical FK percolation with cluster weight $q$ on $D_n$, let $\Gamma_n$ be the collection of all inner and outer boundaries of its open clusters. Let $\Gamma$ be a nested $\mathrm{CLE}_{\kappa'}$ in $D$, where

$$\kappa'=\frac{4\pi}{\arccos(-\sqrt{q}/2)} \in (4,6], \qquad \kappa=\frac{16}{\kappa'} \in [8/3,4).$$

Conformal-invariance conjecture. The collections $\Gamma_n$ converge in distribution to $\Gamma$ with respect to the metric $d_{\mathfrak{L}}$. This conjecture describes the expected scaling limit of FK loop ensembles; it is currently known only for $q=2$, corresponding to $\kappa'=16/3$, and remains open for the other values in the stated range.