The affine hyperplane arrangement conjecture for left cells

Let $\Phi^+$ be the set of positive roots, with short and long roots distinguished, and define

$$\nu\colon \Phi^+\rightarrow \mathbb{Z}_{\geq 0}$$

by $\nu(\alpha)=0$ if $\alpha$ is short and $\nu(\alpha)=1$ if $\alpha$ is long. Let $\mathscr{H}_{\nu}$ be the affine hyperplane arrangement from the paper's defining equation, and let $\mathscr{R}_{\nu}$ be the connected components of $V\smallsetminus\mathscr{H}_{\nu}$, where $V$ is the ambient real vector space. The affine arrangement conjecture. Any left cell of the affine Weyl group $\widetilde{W}$ is a union of regions from $\mathscr{R}_{\nu}$. This conjecture proposes a geometric description of left cells using a reduced affine hyperplane arrangement. The source presents this as a conjecture motivated by examples and proves related regularity results, but gives no resolution of this assertion in the stated generality.