Almkvist's strong Lefschetz conjecture for relative coinvariant rings

Fix positive integers $m,n$, let $W=G(m,1,n)$ and $K=\mathfrak{S}_n\subset W$, and define

$$A(m,n)=R^{\mathfrak{S}_n}_{G(m,1,n)}=\frac{\mathbb{C}[e_1,\ldots,e_n]}{(\hat e_1,\ldots,\hat e_n)}.$$

Here $e_i$ is the $i$-th elementary symmetric function in $x_1,\ldots,x_n$, and $\hat e_i$ is the $i$-th elementary symmetric function in $x_1^m,\ldots,x_n^m$. Almkvist's strong Lefschetz conjecture. For fixed $m$, the graded Artinian complete intersection $A(m,n)$ has the strong Lefschetz property for all sufficiently large $n$. Moreover, $A(m,n)$ has maximum Jordan type compatible with its Hilbert function; in particular, it is strong Lefschetz if and only if its Hilbert function is unimodal, and it is always weak Lefschetz. This is an algebraic strengthening of Almkvist's conjecture on unimodality of the coefficients of the Hilbert polynomial. The source presents these assertions as an extension of Almkvist's conjecture; their general validity remains open.