Conjectured scaling limit of critical random-cluster interfaces

Let $q$ satisfy $0\le q\le 4$, set $p=p_c(q)$, and let $(\Omega^{\delta};x^{\delta},y^{\delta})$ be a sequence of discrete Dobrushin domains converging in the Carathéodory sense to a Dobrushin domain $(\Omega;x,y)$. The interface is the exploration path in the loop representation of the random-cluster model, with cluster weight $q$ and Dobrushin boundary conditions.

Critical random-cluster interface conjecture. As $\delta\to 0$, the interface in $(\Omega^{\delta};x^{\delta},y^{\delta})$ converges weakly to the chordal $\operatorname{SLE}_{\kappa}$ connecting $x$ and $y$, where

$$\kappa=\frac{4\pi}{\arccos(-\sqrt{q}/2)}.$$

This conjecture extends the rigorously established FK-Ising case $q=2$ to all $0\le q\le 4$; the claimed convergence describes the scaling limit of critical random-cluster interfaces.