Cooper–Embree–Ha–Ha–Hoefel lower-bound conjecture for monomial ideals

Let $I$ be a monomial ideal, and let $\operatorname{big-height}(I)$ denote the maximum of the heights of the associated primes $P\in\operatorname{Ass}(I)$. Write $e=\operatorname{big-height}(I)$, and let $\alpha(I)$ and $\widehat\alpha(I)$ denote the initial degree and Waldschmidt constant of $I$, respectively. Cooper–Embree–Ha–Ha–Hoefel's conjecture.

$$\widehat\alpha(I)\geq\frac{\alpha(I)+e-1}{e}.$$

This is a Chudnovsky-like lower bound for the Waldschmidt constant of a monomial ideal, extending the analogous proposed bound for ideals of points in projective space. The supplied text does not indicate whether the conjecture has been resolved.