In the set up of Lusztig's Hecke algebra with unequal parameters, let $W$ be a universal Coxeter group with finite many simple reflections, that is, $W=\langle s_i,i=1,2,\cdots,n | s_i^2 =1\rangle$. Let $L$ be a weight function on $W$ with positive integer values, and $\mathcal H$ the corresponding Hecke algebra. Denote $S$ as the set of simple reflections, i.e., $S =\{s_1, \cdots s_n\}$. There is a map $\mathbf a':W \rightarrow \mathbb N$, sending $z \in W$ to $\max _{s \in S, s \leq z} L(s)$. For $y\leq w$ in $W$, $p_{y,w}$ is the newly defined Laurent polynomial, called the Kazhdan-Lusztig polynomial.

I am wondering if it is true that

$\deg p_{y,w} \leq \mathbf a'(y)-\mathbf a'(w).$

Thankful for any proof, hint or counterexample.


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