Let $(W, S)$ be a Coxeter system, and let $T = \bigcup_{w \in W, s \in S} wsw^{-1}$. Associated to every element $t \in T$ is a unique positive root $\alpha_t \in \Phi^{+}$ considered as a vector in the standard geometric representation $V$ of $W$. A total order on $T$ is a **reflection order** if, whenever $\alpha_{t_1} < \alpha_{t_2}$, it follows that $\alpha_{t_1} < x \alpha_{t_1} + y \alpha_{t_2} < \alpha_{t_2}$ whenever the middle term is a positive root with $x > 0, y > 0$. (See, for example, Bjorner and Brenti's book.)

Fix a reflection order and let $[u, v]$ be a Bruhat interval. A maximal chain $u = w_0 \to w_1 \to ... \to w_m = v$ in the Bruhat order is what I'll call monotonic if $w_i w_{i-1}^{-1} > w_{i+1} w_i^{-1}$ in the reflection order.

There is a nonrecursive formula for the Kazhdan-Lusztig polynomials $P_{u,v}(q)$ which implies that $P_{u,v}(0)$ is equal to the number of monotonic maximal chains in $[u, v]$. This number is known by other means to be equal to $1$, so I know that there is a unique monotonic maximal chain; however, I can't prove this directly. So far all I've been able to do is use the greedy algorithm to prove that at least one monotonic maximal chain exists.

Does anyone have a direct proof of this fact?

**Edit:** No progress, but now I have a more general conjecture which I no longer know by other means is true. Fix a sequence $a_1, ... a_m$ of odd positive integers such that $\sum_i a_i = \ell(v) - \ell(u)$. Then there exists at most one monotonic chain (not necessarily maximal) such that $w_i w_{i-1}^{-1} \in T$ and such that $\ell(w_{i-1}) - \ell(w_i) = a_i$.