Characteristic-two dimension conjecture for 2-spec matrix spaces

Let $\mathbb{K}$ be a field of characteristic $2$ with more than $2$ elements, and let $\mathcal{V}$ be a $2$-spec subspace of $\operatorname{M}_n(\mathbb{K})$. Characteristic-two 2-spec dimension conjecture. If $n=3$ or $n\geq 5$, then

$$\dim \mathcal{V}\leq \binom{n}{2}+3.$$

If $n=4$, then

$$\dim \mathcal{V}\leq \binom{n}{2}+4.$$

The characteristic-not-$2$ analogue with bound $\binom{n}{2}+2$ is proved earlier in the paper, while examples show that characteristic $2$ requires the larger bounds above; the two-element field is excluded because every linear subspace is a $2$-spec space.