The orbit-interval conjecture for automorphisms of Bruhat graphs

Let $w \in \mathfrak{S}_n$ and let $\mathcal{O}=\\{\varphi(e) \mid \varphi \in \operatorname{Aut}(\Gamma(e,w))\\}$ be the orbit of the identity under graph automorphisms of $\Gamma(e,w)$. Orbit-interval conjecture. There is some $v \leq w$ such that

$$\mathcal{O}=[e,v].$$

This conjecture gives a conceptual bridge between self-dual Bruhat intervals and vertex-transitive undirected Bruhat graphs. Its status is not resolved in the supplied text.