Reducedness conjecture for the nilpotent local-model scheme

Fix integers $d$, $r$, and $e$, and let $k$ be the residue field. Consider the closed subscheme $\overline N$ of ${\rm Mat}_{r\times r}$ over $k$ defined by

$$\overline N=\{ A\in {\rm Mat}_{r\times r};\ A^e=0,\ {\rm \det}(T\cdot I-A)\equiv T^r\}.$$

Reducedness conjecture. The scheme $\overline N$ is reduced. This reducedness assertion is used to identify the special fiber of the canonical flat model and to establish its normality and flatness; no resolution is given in the supplied text.