Hiss–Lübeck–Malle conjecture on Brauer trees of principal blocks

Let $\Gamma^\bullet$ denote the graph obtained from the Brauer tree of the principal $\ell$-block by removing the exceptional node and all edges incident to it. For each root of unity $\zeta\in K$, let $m_\zeta$ and $M_\zeta$ index the corresponding Harish–Chandra series, whose characters are labeled $\chi_{m_\zeta},\ldots,\chi_{M_\zeta}$. Hiss–Lübeck–Malle conjecture. The connected components of $\Gamma^\bullet$ are labeled by the Harish–Chandra series, hence by roots of unity $\zeta\in K$; the component corresponding to $\zeta$ is the path

$$\chi_{m_\zeta}—\chi_{m_\zeta+1}—\chi_{m_\zeta+2}—\cdots—\chi_{M_\zeta-1}—\chi_{M_\zeta};$$

and the vertices labeled by $\chi_{m_\zeta}$ are the only nodes connected to the exceptional node. This conjecture predicts that the cohomology of the Deligne–Lusztig variety determines the Brauer tree of the principal $\ell$-block, refining the parametrization of its characters by cohomological data. Its status is not resolved in the supplied source.