Disjunction conjecture for Deligne–Lusztig cohomology

Let $W$ be a Weyl group, let $\phi$ be the Frobenius datum, and let $w\in W$ satisfy $(\lambda(w)\phi)^m=\pi\phi^m$ for some $m\geq1$. For $i\neq j$, consider the compactly supported cohomology representations $H_c^i(X(w),\overline{\mathbf Q}_\ell)$ and $H_c^j(X(w),\overline{\mathbf Q}_\ell)$ of $G$.

Disjunction conjecture for Deligne–Lusztig cohomology.

$$\operatorname{Hom}_{\overline{\mathbf Q}_\ell G}\bigl(H_c^i(X(w),\overline{\mathbf Q}_\ell),H_c^j(X(w),\overline{\mathbf Q}_\ell)\bigr)=0.$$

This predicts that distinct cohomological degrees have no common direct representation-theoretic contribution. It is known for Coxeter elements, groups of rank $2$, and in general for $\operatorname{GL}_n$, but remains open in general.