OD-characterizability conjecture for automorphism groups of binary linear groups

Let $L_n(2)$ denote the projective special linear group over the binary field, and let ${\rm Aut}(L_n(2))$ be its automorphism group, for an integer $n\geqslant 2$. A finite group is OD-characterizable if it is uniquely determined, up to isomorphism, by its order and degree pattern. Automorphism-group OD-characterizability conjecture. The groups ${\rm Aut}(L_n(2))$ are OD-characterizable for all integers $n\geqslant 2$. This extends the corresponding recognition question from the groups $L_n(2)$ to their automorphism groups. The source records several small cases and proves the result for ${\rm Aut}(L_p(2))$ and ${\rm Aut}(L_{p+1}(2))$ when $2^p-1$ is a Mersenne prime, but leaves the all-$n$ assertion as a conjecture in the provided text.