Hochster–Seibert–Schenck conjecture on powers of general linear forms

Let $R=k[x_1,\dots,x_r]$, let $L_1,\dots,L_n$ be generic linear forms in $R_1$, and set

$$I=\langle L_1^t,\dots,L_n^t\rangle.$$

Assume $n\geq r+1\geq5$. The quotient $R/I$ is said to have the weak Lefschetz property (WLP) when multiplication by a general linear form has maximal rank in every degree.

Hochster–Seibert–Schenck conjecture. The quotient $R/I$ fails the WLP for all sufficiently large values of $t$.

The conjecture concerns powers of generic linear forms and is one of the open problems highlighted after known results in even and seven variables. It is supported by computational evidence, but the source gives no proof in the stated generality.