Automorphism conjecture for connected complexes of regions

Let $S_g$ be a closed orientable surface and let $\mathcal C_A(S_g)$ be a complex of regions associated to $S_g$. Assume that $\mathcal C_A(S_g)$ is connected and admits no exchange automorphisms. Automorphism conjecture for complexes of regions. The natural map

$$\operatorname{Mod}^{\pm}(S_g)\longrightarrow \operatorname{Aut}\mathcal C_A(S_g)$$

is an isomorphism. This is the stronger statement proposed in the source beyond the theorem requiring a small element; its resolution is not given.