Chudnovsky bound for monomial ideals

Let $I$ be a monomial ideal. Define its initial degree by $[?]$ and its big-height by

$$\operatorname{big-height}(I)=\max\{\operatorname{ht}(P)\mid P\in\operatorname{Ass}(I)\}.$$

The Chudnovsky bound. The Waldschmidt constant satisfies

$$\widehat{\alpha}(I)\geq \frac{\alpha(I)+\operatorname{big-height}(I)-1}{\operatorname{big-height}(I)}.$$

This conjecture strengthens the Skoda bound $\widehat{\alpha}(I)\geq \alpha(I)/\operatorname{big-height}(I)$ and was formulated in the cited work; its status is not resolved in the supplied source.