Lower-bound conjecture for the Waldschmidt constant of fat point schemes

Let $Z\subset {\mathbb{P}}^N$ be a fat point scheme. For a homogeneous ideal $J$, write $\alpha(J)$ for its least nonzero degree, and write $\widehat{\alpha}(I(Z))$ for the Waldschmidt constant of $I(Z)$. Waldschmidt-constant lower-bound conjecture. For every integer $m\geq 1$, one should have

$$\widehat{\alpha}(I(Z))\geq \frac{\alpha(I(mZ))+N-1}{m+N-1}.$$

The paper explains that this inequality would follow from a proposed refinement of the Harbourne–Huneke containment, but the inequality itself is presented as a conjectural consequence and remains open in the stated generality.