The irreducibility conjecture for higher-level Deligne–Lusztig cohomology

Let $X_h$ be the closed subscheme of $U^{n,q}_h$ defined over ${\mathbb F}_{q^n}$, with commuting left action of $U^1_L/U^h_L$ and right action of $U^{n,q}_h({\mathbb F}_{q^n})$. For a character $\chi:U^1_L/U^h_L\to\overline{{\mathbb Q}}_\ell^\times$, write $H^i_c(X_h,\overline{{\mathbb Q}}_\ell)[\chi]$ for the $\chi$-isotypic subspace of compactly supported cohomology. The irreducibility conjecture. There exists $r\geq 0$ such that

$$H^i_c(X_h,\overline{{\mathbb Q}}_\ell)[\chi]=0$$

for all $i\neq r$, and $H^r_c(X_h,\overline{{\mathbb Q}}_\ell)[\chi]$ is an irreducible representation of $U^{n,q}_h({\mathbb F}_{q^n})$. This predicts concentration in a single cohomological degree and irreducibility of every character-isotypic representation; no general resolution is given in the source.