Harbourne–Schenck–Seceleanu conjecture on the WLP for powers of general linear forms

Let $r,n,d$ be positive integers, let $\ell_1,\ldots,\ell_r$ be general linear forms in $\Bbbk[x_1,\ldots,x_n]$, and let $\Bbbk$ be a field of characteristic zero. Define

$$R_{n,r,d}=\Bbbk[x_1,x_2,\ldots,x_n]/\langle \ell_1^d,\ell_2^d,\ldots,\ell_r^d\rangle.$$

A graded algebra has the Weak Lefschetz property (WLP) if multiplication by some linear form has maximal rank in every degree.

Harbourne–Schenck–Seceleanu conjecture. If $r+1\geq n\geq5$, then $R_{n,r,d}$ fails the WLP for $d\gg0$.

This is a more general but less precise conjecture than the odd uniform almost-complete-intersection conjecture above. The source does not state a resolution of this broader asymptotic claim.