Ashkin–Teller magnetisation-interface percolation conjecture

Consider the clusters in the decomposition of the Ashkin–Teller magnetisation field under $+/+$ boundary conditions. Assign independently to each cluster a fair random sign, except that the boundary cluster is assigned $+1$. Clusters with the same sign that touch are merged into a single cluster; call the resulting positive and negative clusters $\oplus$-clusters and $\ominus$-clusters, respectively.

Ashkin–Teller magnetisation-interface percolation conjecture. The law of the outermost interfaces of the Ashkin–Teller magnetisation field is the law of the outermost outer boundaries of the $\ominus$-clusters. The complete collection of interfaces is obtained by iterating this construction.

This is the proposed continuum percolation description of the scaling limit of magnetisation-field interfaces, extending the discrete representation through the dual complement of the Ashkin–Teller current. Even at $U=0$, it suggests a connection between $\operatorname{CLE}_3$ and Gaussian-free-field two-valued sets; the conjecture remains open.