The maximal-rank conjecture for elementary symmetric polynomials

Let $n\geq 3$, $d\leq n$, and let

$$A=\mathbf{k}[x_1,\dots,x_n]/(x_1^d,\dots,x_n^d).$$

The elementary symmetric polynomial of degree $d$ is

$$e_d=\sum_{1\leq i_1<i_2<\cdots<i_d\leq n}x_{i_1}\cdots x_{i_d}.$$

Maximal-rank conjecture for $e_d$. The element $e_d$ is a maximal rank element on $A$ if and only if either $d=2$, or $d=3$ and $n\not\equiv 2\pmod{3}$.

The classification is conjectural in three or more variables; the stated conditions are known to be necessary, while sufficiency remains open in general.