Let $(W,S)$ be a Coxeter system. For any subset $I\subseteq S$, we can have the *parabolic Kazhdan-Lusztig polynomial* $P_{x,w}^I(q)$ with respect to $I$.

Now consider $I\subseteq J\subseteq S$. Both $(W,S)$, $(W_J,J)$ are Coxeter systems.

Since $I\subseteq J$, we get the *parabolic Kazhdan-Lusztig polynomial* $\overline{P}_{x,w}^I(q)$ with respect to $I$ when considering the Coxeter system $(W_J,J)$.

Does $P_{x,w}^I(q)=\overline{P}_{x,w}^I(q)$ for all $x,w\in W_J$?