The Drinfeld stratification cohomology conjecture

Let $\mathbf{G}$ be a reductive group with twisted Levi subgroup $\mathbf{L}$ containing $\mathbf{S}$, and let $R_{0+}$ and $\mathbf{G}^{0}(\mathbf{S},\theta)$ be as in the Drinfeld stratification. Let $X_r$ and $X_r^{(\mathbf{L})}$ be the associated schemes, and let $H_c^{\ast}(-,\overline{\mathbb Q}_\ell)[\theta]$ denote the compactly supported cohomology isotypic component for the character $\theta$.

Drinfeld stratification conjecture. If $R_{0+}\subset R(\mathbf{S},\mathbf{L})$ (or, equivalently, $\mathbf{G}^{0}(\mathbf{S},\theta)\subset\mathbf{L}$), then

$$H_c^{\ast}(X_r, \overline{\mathbb Q}_\ell)[\theta] \cong H_c^{\ast}(X_r^{(\mathbf{L})}, \overline{\mathbb Q}_\ell)[\theta].$$

This conjecture concerns the cohomology of the Drinfeld stratification and was previously known as a conjecture for $\operatorname{GL}_n$. The paper proves part of it, while the special case $\mathbf{L}=\mathbf{S}$ was already proved using geometric techniques; therefore the conjecture as stated is solved.