Permutation-representation conjecture for local systems on parabolic Lusztig varieties

For $z\in {}^J S_n$, define $J_z:=\bigcap_{m\in\mathbb{Z}}z^mJz^{-m}$ and $(S_n)_{J_z}^z:=\\{w\in (S_n)_{J_z}:zw=wz\\}$, where $(S_n)_{J_z}$ is generated by the simple transpositions in $J_z$. Let $L_{z,w}^J$ be the local systems appearing in the decomposition

$$f_*(IC_{\mathcal{Y}_w})=\bigoplus_{z\in {}^J W}IC_{\mathcal{Y}_{z,J}}(L_{z,w}^J).$$

Permutation-representation conjecture. The local systems $L_{z,w}^J$ are induced by permutation representations of $(S_n)_{J_z}^z$. This would strengthen the preceding theorem, which only establishes induction from representations of this group; it is presented as an extension of the Stanley--Stembridge and Haiman conjectures.