Schmutz Schaller's automorphism conjecture for the Schmutz Schaller curve graph

Let $S$ be the surface under consideration, and let $\mathcal{SC}(S)$ denote the Schmutz Schaller curve graph, with vertices and edges as defined in the paper. Let $\operatorname{Mod}^{\pm}(S)$ be the extended mapping class group of $S$. Schmutz Schaller's conjecture. The automorphism group of $\mathcal{SC}(S)$ is isomorphic to

$$\operatorname{Mod}^{\pm}(S).$$

Schmutz Schaller proposed this conjecture for the automorphism group of the curve graph, and the paper resolves it for all but the cases $(g,p)=(1,3)$ and $(g,p)=(0,5)$, so the general statement remains open in those exceptional cases.