The evolutionary initial-degree conjecture for point ideals

Let $K[\mathbb{P}^{N}]$ be the homogeneous coordinate ring, let $I$ be the ideal of a finite set of points in $\mathbb{P}^{N}$, and let $\alpha(I)$ denote its initial degree. Evolutionary initial-degree conjecture. For every $r>0$, one has

$$\alpha\bigl(I^{(rN-(N-1))}\bigr)\geq r\alpha(I)+(r-1)(N-1).$$

This inequality is a necessary consequence of the evolutionary containment conjecture and gives a numerical refinement of the usual Waldschmidt bounds. The paper notes that it holds in elementary cases such as $r=1$ or $\alpha(I)=1$, but leaves it open in general.