The maximal-rank conjecture for complete homogeneous symmetric polynomials

Let $n\geq 3$ and let

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

where

$$h_d=\sum_{a_1+\cdots+a_n=d}x_1^{a_1}\cdots x_n^{a_n}$$

is the complete homogeneous symmetric polynomial of degree $d$.

Maximal-rank conjecture for $h_d$. The element $h_d$ is a maximal rank element on $A$ if and only if either $d=2$, $d=3$ and $n\not\equiv 1\pmod{3}$, or $d=4$ and $n=3$.

A full classification is not known in three or more variables. The conjecture gives the proposed classification, supported by partial results, of when $h_d$ has maximal rank.