Equivariant Gordan's lemma for symmetric cones

Let $C\subseteq\mathbb{R}^{(\mathbb{N})}$ be a \operatorname{Sym}-equivariantly finitely generated rational cone. Set

$$M=C\cap\mathbb{Z}^{(\mathbb{N})}.$$

Equivariant Gordan's lemma. The monoid $M$ is a $\operatorname{Sym}$-equivariantly finitely generated normal monoid.

This is an equivariant analogue of Gordan's lemma for infinite-dimensional cones invariant under the infinite symmetric group. The paper's abstract says that it gives a full proof of this conjecture, so the claim is resolved.