Fortuin–Kasteleyn SLE scaling-limit conjecture

Let $q\in[0,4]$, and consider the Fortuin–Kasteleyn random-cluster model on the square lattice at the self-dual value $p_{\mathrm{sd}}=\sqrt q/(\sqrt q+1)$, with Dobrushin boundary conditions producing an interface from $a$ to $b$. Fortuin–Kasteleyn SLE scaling-limit conjecture. As the lattice step goes to zero, the law of the interface converges to Schramm–Loewner Evolution with

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

This predicts conformally invariant scaling limits for the FK interfaces. The source presents the claim as conjectural, and its general validity across $q\in[0,4]$ remains open.