Edelman–Reiner conjecture on freeness of extended Catalan and Shi arrangements

Let $V=\mathbb{R}^{\ell}$ be an $\ell$-dimensional Euclidean space, let $\Phi$ be a crystallographic irreducible root system with positive roots $\Phi^+$, exponents $(e_1,\ldots,e_{\ell})$, and Coxeter number $h$. For integers $p\leq q$, define

$$\mathcal{A}(\Phi^+)^{[p,q]}=\{H_{\alpha,k}\mid \alpha\in\Phi^+,\ k\in\{p,p+1,\ldots,q\}\}$$

and let $\mathbf{c}\mathcal{A}(\Phi^+)^{[p,q]}$ denote its cone in $\mathbb{R}\times V$.

Edelman–Reiner conjecture. The following arrangements are free:

1. For $m\geq 0$, the cone $\mathbf{c}\mathcal{A}(\Phi^+)^{[-m,m]}$ of the extended Catalan arrangement has exponents

$$(1,e_1+mh,e_2+mh,\ldots,e_{\ell}+mh).$$

2. For $m\geq 1$, the cone $\mathbf{c}\mathcal{A}(\Phi^+)^{[1-m,m]}$ of the extended Shi arrangement has exponents

$$(1,mh,mh,\ldots,mh).$$

These conjectures concern uniform freeness and explicit exponents for the cones of extended Catalan and Shi arrangements associated with crystallographic irreducible root systems; the source presents them as a conjecture posed by Edelman and Reiner.