Lusztig's cohomological realization conjecture for reductive groups over local rings

Let $G$ be a reductive group with an $\mathbb{F}_{q}$-rational Frobenius morphism $F$, let $G_r$ be the corresponding quotient group, and let $U_r$ be the image in $G_r$ of the unipotent radical of an $F$-stable Borel subgroup. For $xin G_r$, define

$$X_x=\{g\in G_r\mid g^{-1}F(g)\in xU_r\}.$$

The finite group $G_r^F$ acts on $X_x$ by left multiplication, and hence on its $l$-adic compactly supported cohomology.

Lusztig's cohomological realization conjecture. Every irreducible representation of $G_r^F$ appears in the virtual representation

$$\sum_{i\geq 0}(-1)^iH_c^i(X_x,\overline{\mathbb{Q}}_l)$$

for some $x\in G_r$.

For $r=1$, the analogous assertion for the Deligne–Lusztig varieties is known, whereas the cited work notes that the simpler formulation using Weyl-group elements does not suffice for $r\geq 2$. The statement is presented as Lusztig's proposed extension to reductive groups over quotients of local rings; its resolution status is not specified in the supplied text.