Vinberg's finite-generation conjecture for essential signatures

Let $\Sigma=\Sigma_{\mathfrak{g}}$ be the semigroup of essential signatures in $\mathfrak{t}^{*}_{\mathbb{Z}}\oplus\mathbb{Z}^{N}$.

Vinberg's finite-generation conjecture. The semigroup $\Sigma$ is finitely generated.

This is presented as a weaker version of Vinberg's conjecture that $\Sigma$ is generated by the essential signatures of fundamental highest weights. Finite generation ensures that the associated rational cone is defined by finitely many linear inequalities.