The saturation conjecture for toric varieties

Let $X_\Sigma$ be the toric variety and let $f_1,\ldots,f_k$ satisfy Assumption~. Let $Y\subseteq\mathbb{C}^n$ be the affine variety constructed from these polynomials as above. Its prime vanishing ideal is denoted by $I(Y)$. Saturation conjecture. The ideal $I(Y)$ is generated by the $k$ polynomials

$$f_i^h(y)-1,\qquad i=1,\ldots,k.$$

Equivalently, the saturation in Corollary~ is unnecessary. The conjecture is motivated by families of examples in which the saturation is unnecessary, as well as by all examples tested by the authors; its general validity remains open.