Characteristic-two classification conjecture for maximal 1-star-spec spaces

Let $\mathbb{K}$ be a field of characteristic $2$ with more than $2$ elements. Let $\mathcal{V}$ be a $\overline{1}^\star$-spec subspace of $\operatorname{M}_n(\mathbb{K})$ with

$$\dim \mathcal{V}=\binom{n}{2}+2.$$

Write $\operatorname{NT}_r(\mathbb{K})$ for the strictly upper-triangular $r\times r$ matrices, write $\mathfrak{sl}_2(\mathbb{K})$ for the trace-zero $2\times2$ matrices, and let $\vee$ denote the block upper-triangular join. Characteristic-two maximal 1-star-spec classification conjecture. There exists a unique $p\in\{0,\ldots,n-2\}$ such that

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

This is a classification conjecture for the extremal spaces corresponding to the proposed characteristic-two bound; uniqueness is asserted only for the integer $p$, up to similarity of matrix spaces.