According to the increaing-Bruhat-path-explanation of Kazhdan-Lusztig $R$-polynomials(in fact $\tilde{R}$-polynomials), and the fact that $[q^{l(x,y)}]\tilde{R}_{x,y}(q)=1$, there is a unique increasing maximal path in any Bruhat interval under any reflection order. Is there a direct explanation?

Thank Professor Woo for suggesting providing more details. So the following is an detailed description of the increaing-Bruhat-path-explanation of Kazhdan-Lusztig $R$-polynomials, which is from Bjorner and Brenti's book <Combinatorics of Coxeter Groups>, GTM 231, pp.136-144.

Suppose $(W,S)$ is a Coxeter system with $\Phi$ its root system. A total ordering $<$ on $\Phi^+$ is called \textbf{a reflection ordering}, if for all $\alpha,\beta \in \Phi^+$ and $\lambda, \mu >0$ such that $\lambda\alpha+\mu\beta\in \Phi^+$, then $\alpha<\lambda\alpha+\mu\beta<\beta$ or $\alpha>\lambda\alpha+\mu\beta>\beta$.

Given a Bruhat path $\Delta=(a_0,a_1,\cdots,a_r)$ and a reflection ordering $<$, define $l(q)=r$, $D(\Delta;<)=\{i \in [r-1]: a_{i-1}^{-1}a_i>a_{i}^{-1}a_{i+1}\}$, and $R_<(u,v)=\sum_{\Delta\in B(u,v):D(\Delta,<)=\emptyset}{q^{l(\Delta)}}$, where $u,v\in W$, $B(u,v)$ denotes all Bruhat paths from $u$ to $v$. Then Theorem 5.3.4 in Bjorner and Brenti's book shows that $\tilde{R}_{u,v}(q)=R_<(u,v)$, where $q^{l(u,v)}\tilde{R}_{u,v}(q^{\frac{1}{2}}-q^{-\frac{1}{2}})=R_{u,v}(q)$.