Villain-model level lines with general boundary angles converge to SLE(4,ρ)

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 $T$ with boundary values $\exp(-ia)$ on $\partial\Lambda_n\cap\{x:\operatorname{Re}(x)<0\}$ and $\exp(ia)$ on $\partial\Lambda_n\cap\{x:\operatorname{Re}(x)\geq0\}$. For sufficiently small $a$, let $\eta^{(n)}$ be the level line of the imaginary part of $\psi_n$. General-boundary Villain-interface conjecture. The curves $\eta^{(n)}$ converge in law to an $\mathrm{SLE}_4(\rho)$ with

$$\rho=\frac{a}{\lambda\sqrt{t}}-1.$$

This extends the preceding SLE4 prediction to more general piecewise boundary values. The scaling-limit assertion is not resolved in the supplied text.