Arctic ellipse conjecture for random lozenge tilings

Consider a lozenge tiling of a hexagon with normalized side lengths $\alpha,\beta,\gamma$, and define the arctic region to be the set of lozenges connected to the boundary by sequences of adjacent lozenges of the same orientation, where adjacency means sharing an edge. In normalized coordinates, let the inscribed ellipse be the ellipse inscribed in the hexagon. Arctic ellipse conjecture. For every fixed $\varepsilon>0$, the probability that the boundary of the arctic region lies more than distance $\varepsilon$ from the inscribed ellipse is exponentially small in the scaling factor $\sigma$. This is the lozenge-tiling analogue of the arctic circle theorem for domino tilings of Aztec diamonds. The source gives numerical and analogous evidence, but the assertion remains open there.