The disjointness conjecture for Deligne–Lusztig cohomology degrees

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$ and let ${\bf X}({\bf w})$ be the associated Deligne–Lusztig variety. The compactly supported cohomology groups $H^i_c({\bf X}({\bf w}))$ carry actions of the finite group ${{\bf G}^F}$. Disjointness conjecture. The groups $H^i_c({\bf X}({\bf w}))$ are disjoint from each other as ${{\bf G}^F}$-modules. The conjecture is presented after the character and specialness conjecture and is proved only in some cases described in the paper, so it remains open in general.