Migliore–Miró-Roig–Nagel conjecture on powers of general linear forms

Let $n$ and $d$ be positive integers, let

$$R={\bf K}[x_1,\dots,x_{2n+1}],$$

and let $L\in R$ be a general linear form. Set

$$I=\langle x_1^d,\dots,x_{2n+1}^d,L^d\rangle.$$

Migliore–Miró-Roig–Nagel conjecture. If $n=3$, then $R/I$ fails the WLP if and only if $d\geq 3$; if $n\geq 4$, then $R/I$ fails the WLP if and only if $d>1$. The conjecture concerns the weak Lefschetz property for quotients by powers of general linear forms, and the source provides no resolution of these cases.