Conjecture B on top-degree Coxeter characters

Let $W$ be a finite Coxeter group of rank $n$, let $W_L$ denote a parabolic subgroup generated by a subset $L$ of a Coxeter generating set, and call a conjugacy class cuspidal if it does not meet any proper standard parabolic subgroup. Let $\mathcal C$ be representatives of the cuspidal conjugacy classes, let $\rho_W^n$ be the relevant top-degree constituent of the regular character, and let $\omega_W^n$ be the top-degree Orlik–Solomon character. Conjecture B. For each $w\in\mathcal C$, there exists a linear character $\varphi_w$ of $C_W(w)$ such that

$$\rho_W^n=\sum_{w\in\mathcal C}\operatorname{Ind}_{C_W(w)}^W\varphi_w=\epsilon\omega_W^n.$$

This is the cuspidal, top-degree component of the refined Lehrer–Solomon conjecture. The paper states that it was proved for symmetric and dihedral groups and proves it for all finite Coxeter groups of rank at most four; its general status is not resolved in the source.