The no-genus-bound automorphism conjecture for complexes of regions

Let $S_g$ be a closed surface of genus $g$, and let $\mathcal C_A(S_g)$ be a complex of regions that is connected and has no holes or corks. Complex-of-regions conjecture. The natural map

$$\operatorname{MCG}(S_g)\longrightarrow\operatorname{Aut}\mathcal C_A(S_g)$$

is an isomorphism. This removes the hypothesis $g\geq3\bar g(A)+1$ from the authors' theorem; the excerpt gives no resolution of the conjecture.