Refined flat-facet conjecture for three-dimensional random crystals

Let $\beta^{-1}$ be sufficiently low, and let $\beta$ denote the inverse temperature. For a random crystal $\Gamma(\sigma)$ in the discrete torus $\mathbb{T}_N$, let $L_i=L_i(\sigma)\subset\mathbb{T}_N$, $i=1,\ldots,6$, be six distinct two-dimensional planes, two orthogonal to each coordinate direction; let $\operatorname{diam}$ denote diameter and $\operatorname{Area}$ the number of plaquettes belonging to the indicated plane. For each $i$, let $\mathbf n_i$ be the unit vector orthogonal to $L_i$ and pointing away from $\Gamma(\sigma)$.

Refined flat-facet conjecture. The event that there are such planes for which $L_i\cap\Gamma(\sigma)$ are flat facets in the following sense has probability tending to $1$ as $N\to\infty$: for every $i$,

$$\operatorname{diam}(L_i\cap\Gamma(\sigma))\ge C_1(\beta)\operatorname{diam}(\Gamma(\sigma)),\qquad C_1(\beta)\to\sqrt{2/3}\text{ as }\beta\to\infty;$$

for every $i\ne i_0=i_0(\sigma)$,

$$\frac{\operatorname{Area}(L_i\cap\Gamma(\sigma))}{[\operatorname{diam}(L_i\cap\Gamma(\sigma))]^2}\ge C_2(\beta),\qquad C_2(\beta)\to\frac12\text{ as }\beta\to\infty;$$

and

$$\frac{\operatorname{Area}(L_{i_0}\cap\Gamma(\sigma))+\operatorname{Area}((L_{i_0}+\mathbf n_{i_0})\cap\Gamma(\sigma))}{[\operatorname{diam}(L_{i_0}\cap\Gamma(\sigma))]^2}\ge C_2(\beta).$$

The source presents this refined version as probably correct, after noting that the preceding stronger conjecture is probably wrong. The intended facet shapes are related to the corresponding Wulff construction.