Naive Chudnovsky bound for monomial ideals

Let $I$ be a monomial ideal. Define its initial degree by $\alpha(I)$, its naive Waldschmidt constant by $\widetilde{\alpha}(I)$, and its big-height by

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

The naive Chudnovsky bound. The naive Waldschmidt constant satisfies

$$\widetilde{\alpha}(I)\geq \left\lfloor \frac{\alpha(I)+\operatorname{big-height}(I)-1}{\operatorname{big-height}(I)}\right\rfloor.$$

This is proposed as a counterpart to the Chudnovsky bound for the Waldschmidt constant; the supplied source gives no evidence that it has been resolved.