Cooper–Embree–Hà–Hoefel conjecture on Waldschmidt constants of monomial ideals

Let $I\subset K[x_1,\ldots,x_n]$ be a monomial ideal, and let $h$ be its big height. Let $\alpha(I)$ be the least degree of a nonzero element of $I$, and let

$$\widehat{\alpha}(I)=\lim_{s\to\infty}\frac{\alpha(I^{(s)})}{s}$$

be its Waldschmidt constant. Cooper et al.'s conjecture. One has

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

For ideals of big height $n$, symbolic and ordinary powers coincide, so the bound reduces to the corresponding equality case; the conjecture is proved in the paper for several classes of monomial ideals.