Malara–Migliore–Nagel conjecture on the weak Lefschetz property

Let $K$ be a field of characteristic zero, let $R=K[x_0,\ldots,x_n]$, and let $L\in R$ be a general linear form. Consider the ideal

$$I=(x_0^d,\ldots,x_n^d,L^d).$$

The weak Lefschetz property (WLP) for an Artinian graded algebra $R/I$ means that multiplication by a general linear form has maximal rank in every degree. Malara–Migliore–Nagel's conjecture. If $n\geq8$ is even, then $R/I$ fails the WLP if and only if $d>1$. The conjecture concerns the WLP for almost complete intersections generated by powers of general linear forms. The paper proves an asymptotic version for sufficiently large $n$ and $d$, but does not establish the stated uniform assertion in all even dimensions $n\geq8$.