Villain-model level lines with piecewise boundary values converge to SLE4

Let $\Lambda_n$ be the finite domain, let $T$ be sufficiently small, and let $\psi_n$ be a Villain model in $\Lambda_n$ at temperature $T$. Give it boundary values $\exp(-i\lambda\sqrt{T_{\mathrm{Vil}}'})$ on the left boundary portion $\partial\Lambda_n\cap\{x:\operatorname{Re}(x)<0\}$ and $\exp(i\lambda\sqrt{T_{\mathrm{Vil}}'})$ on the complementary portion $\partial\Lambda_n\cap\{x:\operatorname{Re}(x)\geq0\}$. Let $\eta^{(n)}$ be the level line of the imaginary part of $\psi_n$. Villain-level-line SLE4 conjecture. As $n$ tends to infinity, $\eta^{(n)}$ converges in law to an $\mathrm{SLE}_4$.

This is the Villain-model analogue of the discrete Gaussian-free-field level-line prediction and is motivated by the Fröhlich–Spencer approximation. The supplied text does not prove the claimed scaling limit.