Vinberg's saturation conjecture for essential signatures

Let $\mathfrak{t}_{\mathbb{Z}}$ be the coroot lattice, let $\Sigma=\Sigma_{\mathfrak{g}}\subset\mathfrak{t}^{*}_{\mathbb{Z}}\oplus\mathbb{Z}^{N}$ be the semigroup of essential signatures, and let $\Sigma_{\mathbb{Q}}$ be the rational cone spanned by $\Sigma$.

Vinberg's saturation conjecture. The semigroup $\Sigma$ is saturated, namely

$$\Sigma=\Sigma_{\mathbb{Q}}\bigcap (\mathfrak{t}_{\mathbb{Z}}^{*}\oplus\mathbb{Z}^{N}).$$

The conjecture would imply that the bases of the representations $V(\lambda)$ are parametrized by lattice points in flat sections of a polyhedral cone. The source notes that the related conjectures were proved in cases $B_{3}$ and $D_{4}$, but does not establish the general statement here.