Abelian defect group cohomology conjecture for Deligne–Lusztig varieties

Let $\mathbf X_{\mathbf w\boldsymbol\varphi}$ be the relevant Deligne–Lusztig variety for $\mathbf G^F$, and define

$$H_c^{\mathrm{odd}}=\bigoplus_{i\,\mathrm{odd}}H_c^i(\mathbf X_{\mathbf w\boldsymbol\varphi},\overline{\mathbb Q}_\ell),\qquad H_c^{\mathrm{even}}=\bigoplus_{i\,\mathrm{even}}H_c^i(\mathbf X_{\mathbf w\boldsymbol\varphi},\overline{\mathbb Q}_\ell).$$

Abelian defect group cohomology conjecture. The two $\mathbf G^F$-modules $H_c^{\mathrm{odd}}$ and $H_c^{\mathrm{even}}$ are disjoint, and $F^\delta$ is semisimple on $H_c^i(\mathbf X_{\mathbf w\boldsymbol\varphi},\overline{\mathbb Q}_\ell)$ for every $i\geq 0$.

These are presented as special cases of abelian defect group conjectures for finite reductive groups. The source does not state a general resolution.