Characteristic-two dimension conjecture for 1-star-spec matrix spaces

Let $\mathbb{K}$ be a field of characteristic $2$ with more than $2$ elements, and let $\mathcal{V}$ be a $1^\star$-spec subspace of $\operatorname{M}_n(\mathbb{K})$. Characteristic-two 1-star-spec dimension conjecture. Then

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

For fields of characteristic different from $2$, the paper proves the smaller bound $\binom{n}{2}+1$, and the characteristic-two example discussed immediately before the conjecture shows why an additional dimension is allowed. The assertion remains open in the stated characteristic-two setting.