Regularity conjecture for zero-dimensional subschemes in toric varieties

Let $X$ be the projective variety with Cox ring $S$, let $U\subset X$ be the open set used to define the Cox-ring setup, and let $I=\left\langle f_1,\ldots,f_n\right\rangle\subset S$ be a homogeneous ideal such that $V_X(I)$ is a zero-dimensional subscheme of $U$. Write $\alpha_i=\deg(f_i)\in\operatorname{Pic}(X)$ for the basepoint-free degrees of the generators, and let $\operatorname{Cl}(X)_+$ denote the positive degree semigroup. For a degree $\alpha$, let $\operatorname{Reg}(I)$ be the set of degrees for which the equivalent interpolation and Hilbert-function conditions in the definition of regularity hold. Let $\zeta_j$ be the points associated with the setup, and say that a point is a basepoint of $S_{\alpha_0}$ when every section of degree $\alpha_0$ vanishes there.

Regularity conjecture. Then

$$\alpha_0+\alpha_1+\cdots+\alpha_n\in\operatorname{Reg}(I)$$

for every $\alpha_0\in\operatorname{Cl}(X)_+$ such that no $\zeta_j$ is a basepoint of $S_{\alpha_0}$.

This conjecture predicts a uniform degree in which the Cox-ring interpolation conditions become regular for zero-dimensional schemes, extending the familiar principle that sufficiently large basepoint-free degrees should recover the ideal component and the expected Hilbert function. The paper presents it as motivated by experimental evidence; no resolution is supplied in the given source.