Harbourne–Huneke odd alpha conjecture for fat point schemes

Let $k$ be an algebraically closed field of characteristic $0$, let $\mathbb W$ be a finite set of distinct points in $\mathbb P^n$, and let $\alpha(\mathbb Y)$ denote the least degree of a nonzero form in the ideal of a fat point scheme $\mathbb Y$. For each integer $r\geq 1$, consider the fat point scheme $(rn-(n-1))\mathbb W$. Harbourne–Huneke's odd alpha conjecture.

$$\alpha((rn-(n-1))\mathbb W)\geq r\alpha(\mathbb W)+(r-1)(n-1).$$

This is the initial-degree implication of the second symbolic-power containment conjecture. The paper verifies it for special line-count configurations in $\mathbb P^2$, while the general assertion remains open in the source.