Subpolynomial vertex-stabiliser growth for almost simple tetravalent arc-transitive graphs

Let $c>0$. Let ${\mathcal F}_c$ be a finite family of graphs, let $G$ be an almost simple group, and let $\Gamma$ be a connected tetravalent $G$-arc-transitive graph with vertex $v$. Vertex-stabiliser growth conjecture. If $\Gamma$ is not contained in ${\mathcal F}_c$, then

$$|G_v|<|\mathrm V\Gamma|^c.$$

The graphs $\Delta_m$ provide examples in which the vertex-stabiliser grows faster than any logarithmic function of the number of vertices, but slower than every positive power. The conjecture asserts that, apart from finitely many exceptional graphs for each $c$, this subpolynomial bound holds for all connected tetravalent arc-transitive graphs with almost simple automorphism group.