Cohomological concentration conjecture for 0-toral characters

Let $\mathbb{T}_r$ and $\mathbb{U}_r$ be the groups appearing in the cohomology varieties $Y_{\mathbb{T}_r,\mathbb{U}_r}$, and let $\theta\colon \mathbb{T}_r(\mathbb{F}_q)\to\overline{\mathbb{F}}_\ell^{\times}$ be the reduction of a character $\widetilde{\theta}\colon \mathbb{T}_r(\mathbb{F}_q)\to\overline{\mathbb{Z}}_\ell^{\times}$. The notation $H_c^*(Y_{\mathbb{T}_r,\mathbb{U}_r};\overline{\mathbb{F}}_\ell)_\theta$ denotes the corresponding character component of compactly supported étale cohomology, and a 0-toral character is understood in the sense of Cunningham–Osserman. Cohomological concentration conjecture. If $\theta$ is a 0-toral character, then

$$H_c^*(Y_{\mathbb{T}_r,\mathbb{U}_r};\overline{\mathbb{F}}_\ell)_\theta$$

is non-zero in only one degree. The conjecture extends known concentration results for ordinary Deligne–Lusztig varieties and is proposed in the context of torsion in integral cohomology; its resolution is not stated in the source.