Vinberg's polyhedral-inequality conjecture for essential signatures

Let $\mathfrak{t}_{\mathbb{Z}}$ be the coroot lattice, and let an essential signature of highest weight $\lambda$ be written as $\sigma=(\lambda;p_{1},\dots,p_{N})$.

Vinberg's polyhedral-inequality conjecture. There exist a family of subsets $M_{i}\subset\{1,\dots,N\}$ and a family of elements $l_{i}\in\mathfrak{t}_{\mathbb{Z}}$ such that the set of essential signatures of highest weight $\lambda$ is given by the inequalities

$$\sum_{j\in M_{i}}p_{j}\leq\lambda(l_{i}).$$

This is the claimed polyhedral description of essential signatures and would give a finite system of inequalities when the relevant cone is polyhedral. The supplied text does not state a general resolution; it only records results for related conjectures in types $B_{3}$ and $D_{4}$.