The Fourier pairing conjecture for irreducible spetsial complex reflection groups

Let $W$ be an irreducible spetsial complex reflection group, and write $\operatorname{Irr}(W)$ for its irreducible characters. Let $\operatorname{Deg}_\chi(q)$ and $\operatorname{Feg}_\chi(q)$ denote the generic degree and fake degree associated with $\chi\in\operatorname{Irr}(W)$, and let $h_\chi$ be the corresponding $h$-value. A pairing $\{-,-\}_W$ on $\operatorname{Irr}(W)\times\operatorname{Irr}(W)$ satisfies (T1), (T2), and (T3) when

$$\operatorname{Deg}_\chi(q)=\sum_{\phi\in\operatorname{Irr}(W)}\{\chi,\phi\}_W\operatorname{Feg}_\phi(q),$$ $$\{\chi,\phi\}_W=\{\phi,\chi\}_W,$$

and $\{\chi,\phi\}_W\ne0$ implies $h_\chi=h_\phi$. The Fourier pairing conjecture. For every irreducible spetsial complex reflection group $W$, there exists a pairing $\{-,-\}_W$ satisfying (T1), (T2), and (T3). Such a pairing would provide the Fourier-transform structure motivating the section and would relate generic degrees to fake degrees while respecting the $h$-values; existence for all irreducible spetsial complex reflection groups remains open.