Random-cluster interface convergence to SLE

Let $G_n$ be the partially wired square grid described in the source, let $\omega$ have the critical random-cluster measure with $q\in(0,4)$, let $\omega^\dagger$ be its dual configuration, and let $\beta$ be the locus of points equidistant from the wired primal and dual components. Random-cluster SLE convergence conjecture. As $n\to\infty$, the law of $\beta$ converges weakly, up to reparameterization, to the trace of $\textrm{SLE}_\kappa$, where

$$\kappa=\frac{4\pi}{\cos^{-1}(-\sqrt{q}/2)}.$$

The source presents this as a prediction based on conjectured random-cluster perimeter dimensions and gives no proof.