Lapid's conjecture on parabolic Kazhdan--Lusztig polynomials

Let $P_{u,w}$ be the Kazhdan--Lusztig polynomials for symmetric groups, and let $H\simeq S_m\times\dots\times S_m$ be the parabolic subgroup of $S_{mn}$ of type $(m,\dots,m)$. For $w\in S_n$, let $\widetilde w\in S_{mn}$ be defined by $\widetilde w(mi-j)=mw(i)-j$ for $i=1,\dots,n$ and $j=0,\dots,m-1$. Write

$$\sum_{u\in H}\operatorname{sgn} u\,P_{\widetilde xu,\widetilde w}=q^{{m\choose 2}(\ell(w)-\ell(x))}\widetilde P^{(m)}_{x,w}.$$

For any $x,w\in S_n$ with $x\le w$, Lapid's conjecture. The polynomial $\widetilde P^{(m)}_{x,w}$ satisfies all of the following: it is a polynomial rather than a Laurent polynomial; $\widetilde P^{(m)}_{x,w}(0)=1$; $\widetilde P^{(m)}_{x,w}=\widetilde P^{(m)}_{xs,w}$ for every simple reflection $s$ of $S_n$ such that $ws<w$; and

$$\deg\widetilde P^{(m)}_{x,w}=m\deg P_{x,w}.$$

In particular, $\widetilde P^{(m)}_{x,w}=1$ if and only if $P_{x,w}=1$. The conjecture was motivated by representation-theoretic irreducibility results and computer calculations concerning parabolic Kazhdan--Lusztig polynomials. It was verified numerically when $nm\le 12$ and proved in the paper for $m=2$ when $w$ is a Coxeter element of $S_n$ or of a parabolic subgroup; the general conjecture remains open in the supplied text.