Order-two essential-weight conjecture for extremal matrices

Let $A$ be an extremal multidimensional matrix of order $2$, and let $K$ be a vertex of the polyhedron of polyplexes in $A$. Write $\operatorname{supp}(K)$ for the support of $K$, and let $k_{\alpha}$ be the corresponding values for $\alpha\in\operatorname{supp}(K)$.

Order-two essential-weight conjecture. The values $k_{\alpha}$, for $\alpha\in\operatorname{supp}(K)$, are essential weights of an optimal hyperplane cover for some multidimensional extremal matrix of order $2$.

The conjecture is intended to provide another description of optimal hyperplane covers for extremal matrices of order $2$. The source supplies no proof or resolution.