Villain-model ALE convergence

Let $\Lambda_n$ be the finite domain and let $\psi_n$ be a Villain model in $\Lambda_n$ at sufficiently small temperature $T$. Let the discrete ALE be the union of the interfaces associated with the imaginary part of $\psi_n$, and let $\mathbb{A}_{-\lambda,\lambda}$ denote the continuum ALE. Villain-ALE conjecture. As $n$ tends to infinity, the discrete ALE associated with the imaginary part of $\psi_n$ converges to $\mathbb{A}_{-\lambda,\lambda}$.

This is a stronger, temperature-dependent analogue of the Gaussian-free-field ALE conjecture. The authors emphasize that the limiting geometry is expected not to depend on the temperature once it is sufficiently small, but the conjectured convergence itself remains open in the supplied text.