The evolutionary Eisenbud–Mazur conjecture for point ideals

Let $K[\mathbb{P}^{N}]$ be the homogeneous coordinate ring, let $M$ be its homogeneous maximal ideal, and let $I$ be the ideal of a finite set of points $p_i\in\mathbb{P}^{N}$. Evolutionary Eisenbud–Mazur conjecture. For every $r\geq1$, one has

$$I^{(rN-(N-1))}\subseteq M^{(r-1)(N-1)}I^r.$$

This refines the uniform symbolic-power containment by retaining a prescribed power of the maximal ideal. The paper proves it for point ideals arising from star configurations and complete intersections, while the general conjecture remains open.