Equivalence conjecture for the two shellability conditions

Let $W$ be a Weyl group, let $x\leq w$, and let $\mathfrak w$ be a reduced word for $w$. Let $\lambda_{x,\mathfrak w}$ be the deletion-index set, and let $\mathscr C^+_{x,\mathfrak w}$ and $\mathscr C^-_{x,\mathfrak w}$ be the maximal chains associated with the two shellability conditions; write $\lambda(\mathscr C)$ for the label sequence of a chain and $^*$ for reversal. Equivalence conjecture. The following are equivalent:

$$\lambda_{x,\mathfrak w}=\lambda(\mathscr C^-_{x,\mathfrak w})^*,$$ $$\lambda(\mathscr C^+_{x,\mathfrak w})=\lambda(\mathscr C^-_{x,\mathfrak w})^*,$$ $$\lambda_{x,\mathfrak w}=\lambda(\mathscr C^+_{x,\mathfrak w}).$$

The equivalence is proposed in the setting of the paper's shellability conditions and is motivated by computer tests; no proof or resolution is supplied in the source.