Characteristic-two classification conjecture for maximal 2-spec spaces

Let $\mathbb{K}$ be a field of characteristic $2$ with more than $2$ elements. Let $\mathcal{V}$ be a $\overline{2}$-spec subspace of $\operatorname{M}_n(\mathbb{K})$, with $n\geq3$. Let $\operatorname{NT}_r(\mathbb{K})$ denote the strictly upper-triangular matrices, let $\mathfrak{sl}_2(\mathbb{K})$ denote the trace-zero $2\times2$ matrices, let $\mathcal{H}(\mathbb{K})$ be the exceptional $7$-dimensional $\overline{1}$-spec subspace of $\operatorname{M}_4(\mathbb{K})$ defined in the paper, and let $\vee$ denote the block upper-triangular join. Characteristic-two maximal 2-spec classification conjecture. The following assertions hold: (a) If $n\notin\{4,6\}$ and $\dim\mathcal{V}=\binom{n}{2}+3$, then there is a unique $p\in\{0,\ldots,n-2\}$ such that

$$\mathcal{V}\simeq \mathbb{K}I_n\oplus\bigl(\operatorname{NT}_p(\mathbb{K})\vee\mathfrak{sl}_2(\mathbb{K})\vee\operatorname{NT}_{n-p-2}(\mathbb{K})\bigr).$$

(b) If $n=4$ and $\dim\mathcal{V}=\binom{n}{2}+4$, then

$$\mathcal{V}\simeq\mathfrak{sl}_2(\mathbb{K})\vee\mathfrak{sl}_2(\mathbb{K}).$$

(c) If $n=6$, exactly one of the following holds: for a unique $p\in\{0,\ldots,4\}$,

$$\mathcal{V}\simeq\mathbb{K}I_6\oplus\bigl(\operatorname{NT}_p(\mathbb{K})\vee\mathfrak{sl}_2(\mathbb{K})\vee\operatorname{NT}_{4-p}(\mathbb{K})\bigr),$$

we have $\mathcal{V}\simeq\mathfrak{sl}_2(\mathbb{K})\vee\mathcal{H}(\mathbb{K})$, or we have

$$\mathcal{V}\simeq\mathcal{H}(\mathbb{K})\vee\mathfrak{sl}_2(\mathbb{K}).$$

The paper presents these as plausible characteristic-two classification conjectures, extending the preceding dimension bounds and accounting for the exceptional $4\times4$ space $\mathcal{H}(\mathbb{K})$; no resolution is supplied in the source.