The alternating-form symmetric-operator equality conjecture

Let $b$ be a non-degenerate alternating bilinear form on an $n$-dimensional vector space $V$ over a field with characteristic not $2$. Set

$$\nu:=\frac{n}{2},$$

and let $\mathcal{V}$ be a nilpotent linear subspace of $\mathcal{S}_b$ with dimension $\nu(n-\nu)$. A maximal partially complete $b$-singular flag is denoted by $\mathcal{F}$, and $\mathcal{N}\mathcal{S}_{b,\mathcal{F}}$ denotes the associated nilpotent subspace.

Alternating-form symmetric-operator equality conjecture. There exists a maximal partially complete $b$-singular flag $\mathcal{F}$ of $V$ such that

$$\mathcal{V}=\mathcal{N}\mathcal{S}_{b,\mathcal{F}}.$$

This is one of the two equality cases not covered by the proved results because $\mathcal{S}_b$ is generally not stable under squares when $b$ is alternating; the conjecture asserts that the corresponding classification still holds.