Projection-volume conjecture for finite reflection groups

Let $\mathcal{W}$ be a finite reflection group acting on $\mathbb{R}^n$, with fundamental chamber $\mathcal{C}$ and characteristic polynomial $\chi(t)$ of its associated reflection arrangement. For a generic point $x$ in $\mathcal{C}$, let $\pi_\mathcal{C}(gx)$ denote the projection onto $\mathcal{C}$, and let $k$ be a dimension. Finite reflection-group projection conjecture. The number of elements $g\in\mathcal{W}$ for which $\pi_\mathcal{C}(gx)$ is $k$-dimensional equals the absolute value of the coefficient of $t^k$ in $\chi(t)$. Consequently, the projection volumes for any fundamental chamber of $\mathcal{W}$ are proportional to the absolute values of the coefficients of $\chi(t)$. This extends the result proved in the paper for reflection groups of types $A_n$, $B_n$, and $D_n$ to all finite reflection groups; its general validity is left as a conjecture.