Brendle–Margalit conjectures on complexes of regions and geometric normal subgroups

Let $\Sigma$ be a surface, let $\mathcal{C}_R(\Sigma)$ be a complex of regions, and let $N$ be a normal subgroup of $\operatorname{Map}^{\pm}(\Sigma)$. Brendle–Margalit conjectures for complexes of regions.

1. The natural map

$$\operatorname{Map}^{\pm}(\Sigma)\to\operatorname{Aut}\mathcal{C}_R(\Sigma)$$

is an isomorphism if and only if $\mathcal{C}_R(\Sigma)$ is connected and admits no exchange automorphisms.

2. The natural map

$$\operatorname{Map}^{\pm}(\Sigma)\to\operatorname{Aut}N$$

is an isomorphism if and only if $N$ contains elements whose supports $L,R\subset\Sigma$ are disjoint and separated by a pair of pants. The first assertion concerns automorphisms of complexes of regions, while the second characterizes geometric normal subgroups through elements with separated supports. The supplied text presents this as a rephrasing and generalization of a Brendle–Margalit conjecture and does not give a resolution of both assertions.