Geramita–Harbourne–Migliore conjecture on powers of star-configuration ideals

Let ${\mathcal A}=\{H_1,\ldots,H_s\}$ be a generic collection of $s\geq n+1$ hyperplanes in $\mathbb P^n$, with defining linear forms $\ell_1,\ldots,\ell_s$. For $1\leq c\leq n$, let

$$V_c({\mathcal A})=\bigcup_{1\leq j_1<\cdots<j_c\leq s}H_{j_1}\cap\cdots\cap H_{j_c}$$

be the star configuration of codimension $c$, and let $I(V_c({\mathcal A}))$ denote its defining ideal, $I(V_c({\mathcal A}))^{(m)}$ its $m$-th symbolic power, and $M=\langle x_0,\ldots,x_n\rangle$ the homogeneous maximal ideal. Geramita–Harbourne–Migliore conjecture. For any $m\geq 1$ one has

$$I(V_c({\mathcal A}))^m=I(V_c({\mathcal A}))^{(m)}\cap I(V_{c+1}({\mathcal A}))^{(2m)}\cap\cdots\cap I(V_n({\mathcal A}))^{((n-c+1)m)}\cap M^{(s-c+1)m}.$$

This conjecture describes ordinary powers of star-configuration ideals through symbolic powers of the nested star configurations and the maximal ideal. Its resolution is not established in the supplied text.