Eight-face conjecture for facets of the Newton polytope

Let $\mathbf{N}_{3,n}$ be the Newton polytope, and let $\gamma_{j_1,j_2,j_3}$ be the generalized positive root

$$\gamma_{j_1,j_2,j_3}=\sum_{t=j_1}^{j_2-2}\alpha_{1,t}+\sum_{t=j_2-1}^{j_3-3}\alpha_{2,t}.$$

Assume that $\{j_1,j_2,j_3\}$ is totally nonfrozen, meaning that no two indices are cyclically adjacent. The generalized positive roots are linear functions on $\mathbb{R}^{2\times(n-3)}$.

Eight-face conjecture. The facet of $\mathbf{N}_{3,n}$ minimizing $\gamma_{j_1,j_2,j_3}$ has exactly eight codimension-$3$ faces combinatorially isomorphic to

$$\mathbf{N}_{3,j_2-j_1+2}\times\mathbf{N}_{3,j_3-j_2+2}\times\mathbf{N}_{3,j_1-j_3+n+2}.$$

These faces correspond to the eight weakly separated collections listed in the source. The claim generalizes the observed three-split structure and remains unproved in the supplied text.