Hiß–Lübeck–Malle conjecture on the shape and embedding of Brauer trees

Let $\Gamma$ be the Brauer tree of the principal $\Phi_h$-block. The characters $\chi_{m_\zeta}$ arise from the Frobenius eigenspaces in the rational $\ell$-adic cohomology, and the exceptional vertex is the vertex corresponding to the non-unipotent characters in the block. Hiß–Lübeck–Malle conjecture. (i) The vertices labelled by $\chi_{m_\zeta}$ are the only nodes connected to the exceptional node in $\Gamma$. (ii) The vertices labelled by $\chi_{m_\zeta}$ are ordered around the exceptional vertex according to increasing values of $m_\zeta$. This gives a precise prediction for the Brauer tree's shape and planar embedding from cohomological Frobenius data. The source presents this as the Hiß–Lübeck–Malle conjecture and the parser supplies no resolution status; the paper's abstract nevertheless states that the authors prove the conjecture in the cases considered.