Boundary regularity conjecture for the Liouville density of the discrete Gaussian free field

Let $D$ be a domain, let $t\in\mathbb R$, and let $\rho^D(t,\mathord\cdot)$ denote the density from Theorem 2.7. Boundary regularity conjecture. For each $t\in\mathbb R$, the function $x\mapsto\rho^D(t,x)$ is bounded and tends to zero as $x$ approaches $\partial D$. In particular, the function admits a continuous extension to all of $\overline D$. The preceding results establish absolute continuity of the relevant measure and an integrability condition for this density, but do not determine its boundary regularity; the conjecture is motivated by the stated asymptotics and simulations.