Pappas–Rapoport reducedness conjecture for a symplectic nilpotent scheme

Let $F$ be a field with characteristic $e 2$, and let $n$ be an integer divisible by $4$. Let $J$ be the $n\times n$ matrix defining the symplectic form, and let $W$ be the scheme of $n\times n$ matrices $X$ over $\operatorname{Spec} F$ satisfying

$$-JX^T J=X,\qquad X^2=0,\qquad \operatorname{char}_X(T)=T^n.$$

Pappas–Rapoport's conjecture. The scheme $W$ is reduced.

This is the equal-parts case of a more general conjecture formulated by Pappas and Rapoport for even $n$, depending on a partition of $n$ into two even parts. The paper proves a special case under additional conditions, while the stated version is not established in the supplied context.