The Evasiveness conjecture for vertex-transitive graphs

Let $X$ be a graph. Write $D_{\infty}$ for the class of graphs that are $k$-dismantlable for some finite $k$; equivalently, $X$ is non-evasive when $X\in D_{\infty}$. A graph is vertex-transitive if its automorphism group acts transitively on its vertices.

Evasiveness conjecture for graphs. If $X\in D_{\infty}$ and $X$ is vertex-transitive, then $X$ is a complete graph.

This is the graph-theoretic restriction of the Evasiveness Conjecture for vertex-homogeneous simplicial complexes to clique complexes. The supplied text gives no evidence that the conjecture has been resolved.