Lehrer–Solomon decomposition conjecture for Whitney homology

Let $W$ be a finite real reflection group acting on $V$, let $\mathcal{O}_X$ be a $W$-orbit of intersection subspaces, and let ${{\operatorname{WH}}}_{\mathcal{O}_X}$ be the corresponding Whitney-homology module. For each representative $v$ of a $W$-conjugacy class contained in $\{w\in W:\operatorname{Fix}_w(V)\in\mathcal{O}_X\}$, let $\operatorname{Z}_W(v)$ be its centralizer and let $\xi_v:\operatorname{Z}_W(v)\to\mathbb{C}^{\times}$ be a degree-one character. Lehrer–Solomon conjecture. There is an isomorphism of $W$-modules

$${{\operatorname{WH}}}_{\mathcal{O}_X}\cong\bigoplus_v\operatorname{Ind}_{\operatorname{Z}_W(v)}^W\xi_v.$$

The conjecture was verified in the source for symmetric groups of type $A_{n-1}$ and for dihedral groups; it is presented as open in general.