Generic powers of linear forms fail the weak Lefschetz property

Let $S={\mathbb K}[x_1,\ldots,x_r]$, let $l_i\in S_1$ be generic linear forms, and set

$$I=\langle l_1^t,\ldots,l_n^t\rangle\subseteq S.$$

Generic-powers WLP conjecture. If $n\ge r+1\ge 5$, then $S/I$ fails the weak Lefschetz property for all $t\gg0$.

This predicts asymptotic failure of the weak Lefschetz property for sufficiently many generic powers of linear forms in at least five variables. The paper presents it as a belief based on partial results and computational evidence; its general status is not resolved here.