Conjecture C for parabolic subgroups of finite Coxeter groups

Let $W$ be a finite Coxeter group, let $W_L$ be a proper parabolic subgroup of rank $r$, and let $\mathcal C_L$ represent the cuspidal conjugacy classes of $W_L$. For each $w\in\mathcal C_L$, let $\varphi_w$ be the linear character of $C_{W_L}(w)$ supplied by Conjecture B, and let $\widetilde{\rho_L^r}$ and $\widetilde{\omega_L^r}$ be extensions to $N_W(W_L)$ of the corresponding characters of $W_L$. Let $\alpha_L$ be the composition of the determinant with restriction to the fixed-point subspace of $W_L$. Conjecture C. Each $\varphi_w$ extends to a linear character $\widetilde{\varphi_w}$ of $C_W(w)$ such that

$$\widetilde{\rho_L^r}=\sum_{w\in\mathcal C_L}\operatorname{Ind}_{C_W(w)}^{N_W(W_L)}\widetilde{\varphi_w}=\epsilon\alpha_L\widetilde{\omega_L^r}.$$

This is the relative parabolic version of Conjecture B and is designed to provide the non-cuspidal characters needed for the refined conjecture. The source gives no general resolution status; it is used as an inductive component in the paper’s rank-three and rank-four computations.