Postnikov–Stanley Riemann hypothesis for characteristic polynomials of truncated affine Weyl arrangements

Let $\Phi$ be a root system of rank $\ell$ with Coxeter number $h$. For integers $a,b$ with $a\leq 1\leq b$, let $\mathcal{A}_{\Phi}^{[a,b]}$ be the truncated affine Weyl arrangement

$$\mathcal{A}_{\Phi}^{[a,b]}=\{H_{\alpha,k}\mid \alpha\in\Phi^+,\ k\in\mathbb{Z},\ a\leq k\leq b\}.$$

Let $\chi(\mathcal{A}_{\Phi}^{[a,b]},t)$ denote its characteristic polynomial. Postnikov–Stanley’s Riemann hypothesis. If $a+b\geq 1$, then every root $t\in\mathbb{C}$ of

$$\chi(\mathcal{A}_{\Phi}^{[a,b]},t)=0$$

satisfies

$$\operatorname{Re}t=\frac{h(b-a+1)}{2}.$$

The conjecture asserts that all characteristic-polynomial roots lie on a vertical line, analogous to a Riemann-hypothesis phenomenon. The source states that it was proved by Stanley, Postnikov, and Athanasiadis for $\Phi\in\{A_\ell,B_\ell,C_\ell,D_\ell,G_2\}$, while no general resolution is supplied here.