The Lehrer–Solomon conjecture for finite Coxeter groups

Let $W$ be a finite Coxeter group generated by a set $S$ of simple reflections. Let $\Phi_S$ and $\Psi_S$ denote the characters of the top components of the group algebra $\mathbb{C}W$ and the Orlik–Solomon algebra $A(W)$, respectively. Let $\mathcal{C}$ be the set of cuspidal conjugacy classes of $W$, and choose $w_C\in C$ for each $C\in\mathcal{C}$. Lehrer–Solomon conjecture. For each class $C\in\mathcal{C}$, there exist linear characters $\varphi_{w_C}$ and $\psi_{w_C}$ of $C_W(w_C)$ such that

$$\Phi_S=\sum_{C\in\mathcal{C}}\operatorname{Ind}_{C_W(w_C)}^W\varphi_{w_C},$$ $$\Psi_S=\sum_{C\in\mathcal{C}}\operatorname{Ind}_{C_W(w_C)}^W\psi_{w_C},$$

and $\psi_{w_C}=\varphi_{w_C}\epsilon$ for all $C\in\mathcal{C}$, where $\epsilon$ is the sign character of $W$. This conjecture relates the top components of the group algebra and Orlik–Solomon algebra through induced linear characters of centralizers; it is known for symmetric groups and, according to the paper, is proved by the inductive approach for finite Coxeter groups of rank at most $2$, while the general case remains open.