The toric integral-model Hilbert property conjecture

Let $(K,C)$ be a PF field, let $(X,M)$ be a toric pair, and let $(\mathcal{X},\mathcal{M})$ be its toric integral model. Write $N_M$ and $N_M^+$ for the associated subsets of the cocharacter lattice $N$. For an open subset $B\subsetneq C$, consider the $\mathcal{M}$-Hilbert property over $B$.

Toric integral-model Hilbert property conjecture. If $N_M=N$, then $(\mathcal{X},\mathcal{M})$ satisfies the $\mathcal{M}$-Hilbert property over every open $B\subsetneq C$. If furthermore $N_M^+=N$, then $(\mathcal{X},\mathcal{M})$ satisfies the $\mathcal{M}$-Hilbert property over $C$.

This conjecture proposes that, for split toric varieties, the stated lattice conditions ensure the $\mathcal{M}$-Hilbert property for the toric integral model. The supplied text does not indicate whether either assertion is known or resolved.