$\def\SYT{\mathrm{SYT}}\def\RSK{\mathrm{RSK}}\DeclareMathOperator\evac{evac}$Let $\mathfrak{S}_n$ be the symmetric group, $\SYT_n$ be the set of standard young tableaux of size $n$.

For $u\in \mathfrak{S}_n$, let $\RSK:\mathfrak{S}_n\to \SYT_n^2$ denote the Robinson-Schensted-Knuth correspondance.

Let $P_{u,w}(q)$ be the Kazhdan-Lusztig polynomial and $\mu_{u,w}=[q^{{(l(w)-l(u)-1)}/{2} }]P_{u,w}(q)$.

Let $\evac:\SYT_n\to \SYT_n$ be Schutzenberger's involution.

**Question:** I am looking for a proof of the following proposition that Schutzenberger's involution preserves the $\mu$-coefficient:

Take $u,w\in \mathfrak{S}_n$, and let $(P_1,Q_1)=\RSK(u)$, $(P_2,Q_2)=\RSK(w)$. If we set \begin{gather*} u'=\RSK^{-1}(\evac(P_1),\evac(Q_1)) \\ w'=\RSK^{-1}(\evac(P_2),\evac(Q_2)), \end{gather*} then we have $$\mu(u,w)=\mu(u',w').$$