Ehrenborg–Morel–Readdy's conjecture on pizza quantities for Coxeter arrangements

Let $\mathcal{H}$ be a Coxeter arrangement on a finite-dimensional inner product space $V$, let $a\in V$, and let $R$ satisfy $\|a\|\leq R$. Assume that the number of hyperplanes of $\mathcal{H}$ is greater than $\dim(V)$ and has parity different from $\dim(V)$. Ehrenborg–Morel–Readdy's conjecture. The pizza quantity $P(\mathcal{H},\mathbb{B}(a,R))$ is zero if and only if the center $a$ lies on one of the hyperplanes of $\mathcal{H}$. This conjecture addresses the parity cases not covered by the known vanishing theorem; the supplied text does not establish the assertion, so its resolution remains open.