Two-generation conjecture for traceless matrices over the binary field

Let $n$ be a positive integer, and let $\operatorname{\mathfrak{sl}}_n(\mathbf{F}_2)$ denote the Lie algebra of traceless $n\times n$ matrices over the field with two elements.

Two-generation conjecture. If $n \neq 4$, then $\operatorname{\mathfrak{sl}}_n(\mathbf{F}_2)$ is $2$-generated.

The authors report that random computer searches found generating pairs for every $n\leq 20$ except $n=4$, motivating the conjecture. The statement concerns the unresolved even-characteristic case; the context provided does not establish it for all $n$ or explain the exceptional case $n=4$.