Brendle–Margalit large-support metaconjecture for normal subgroups

Let $S_g$ be a closed orientable surface, let $N$ be a normal subgroup of $\operatorname{Mod}(S_g)$, and call a subsurface not large if it is contained in a nonseparating subsurface homeomorphic to a proper subsurface of its complement. Say that an element has not-large support when it has a representative supported on such a subsurface. Brendle–Margalit large-support metaconjecture. If $N$ contains an element whose support is not large, then both $\operatorname{Aut}(N)$ and the abstract commensurator group of $N$ are isomorphic to $\operatorname{Mod}^{\pm}(S_g)$. This strengthens the proved theorem using small support by replacing the one-third-genus condition with the not-large condition; the source gives no resolution.