The strengthened orbit-interval conjecture for Bruhat-irreducible permutations

Let $w \in \mathfrak{S}_n$ be Bruhat irreducible, and let $\mathcal{O}$ denote the orbit of $e$ under graph automorphisms of $\Gamma(w)$. Define

$$v(w) \coloneqq w_0(D_L(w))\cdot A_R(w) \cdot w_0(D_L(w)\cap D_R(w)) \cdot A_L(w)\cdot w_0(D_R(w)).$$

Strengthened orbit-interval conjecture. Then $\mathcal{O}=[e,v(w)]$. This is presented as a strengthening of the orbit-interval conjecture; its status is not resolved in the supplied text.