Characteristic pure mapping class group and automorphism conjecture

Let $S$ be a surface whose genus is finite and at least $4$. The groups $\operatorname{PMap}(S)$, $\operatorname{Map}(S)$, and $\operatorname{Map}^\pm(S)$ denote the pure mapping class group, mapping class group, and extended mapping class group, respectively. Characteristic and automorphism conjecture.

1. $\operatorname{PMap}(S)$ is a characteristic subgroup of $\operatorname{Map}(S)$ and $\operatorname{Map}^\pm(S)$.
2. If $S$ is borderless, then

$$\operatorname{Aut}(\operatorname{Map}^\pm(S))=\operatorname{Aut}(\operatorname{Map}(S))=\operatorname{Map}^\pm(S).$$

These conjectures extend the preceding rigidity theorem for pure mapping class groups and would recover, for big mapping class groups, the automorphism description known in the finite-type setting. The source gives no resolution of these assertions.