Let $X$ be a normal toric variety over $\mathbb{C}$ with no torus factors. Let $S$ be the Cox ring of $X$, and $B \subset S$ be the irrelevant ideal. Given an ideal sheaf $\mathcal{I} \subseteq \mathcal{O}_X $, we can define the $S$-module $\Gamma_{\star}(\mathcal{I})$ by
$$ \Gamma_{\star}(\mathcal{I}) = \bigoplus_{\alpha \in \operatorname{Cl} X} \Gamma(X, \mathcal{I}(\alpha)) $$
Then we can define the ideal $I$ corresponding to $\mathcal{I}$ as the image of the standard map
$$ \varphi : \Gamma_{\star}(\mathcal{I}) \to \Gamma_{\star}(\mathcal{O}_X) = S $$ My question is: is the ideal $I$ always $B$-saturated? In other words, is it always true that $I = I: B$? This is definitely true for projective space, and, more generally, for smooth toric varieties. But is it true for singular toric varieties $X$? The problem is that the map $\varphi$ does not have to be injective.