Harbourne, Schenck and Seceleanu's asymptotic weak Lefschetz conjecture

Let $K$ be a field of characteristic $0$, let $R=K[x_1,\dots,x_r]$, and let $\ell_1,\dots,\ell_n$ be general linear forms. For a positive integer $t$, set

$$I=(\ell_1^t,\dots,\ell_n^t)\subset R,$$

and let $A=R/I$. Harbourne, Schenck and Seceleanu's asymptotic conjecture. If $n\geq r+1\geq 5$, then $A$ fails the Weak Lefschetz property for $t\gg0$. The source lists this among open conjectures related to ideals generated by powers of general linear forms; no resolution is supplied here.