Local lozenge probability conjecture for boxed plane partitions

Let $\alpha,\beta,\gamma$ be the normalized side lengths of a hexagon, let $(x,y)$ be a normalized location in this hexagon, and let $\mathcal P_{\alpha,\beta,\gamma}(x,y)$ denote the limiting density of vertical lozenges there. Let $V$ be any open set in the hexagon containing the four points at which $\mathcal P_{\alpha,\beta,\gamma}$ is discontinuous. Local lozenge probability conjecture. As the scaling factor $\sigma\to\infty$, the probability of finding a vertical lozenge at normalized location $(x,y)\notin V$ is $\mathcal P_{\alpha,\beta,\gamma}(x,y)+o(1)$, with the $o(1)$ error bound uniform for $(x,y)\notin V$. This conjecture concerns local statistics that are not determined by the paper's macroscopic averaging theorem; numerical evidence and analogous results for domino tilings support it, but it remains unproved here.