Hiss–Lübeck–Malle-plus conjecture on the planar embedding of the Brauer tree

Let $G$ be a finite reductive group whose principal $\ell$-block has cyclic defect, and let $h_0$ be the relevant order of the roots of unity occurring in the block. Let $q^\delta$ act on the relevant torus and write its reduction modulo $\ell$ as an element of the group of $h_0$-th roots of unity. Hiss–Lübeck–Malle-plus conjecture. The $\ell$-reduction of $q^\delta$ gives a canonical generator of the group of $h_0$-th roots of unity, and the corresponding order induces the cyclic ordering around the exceptional node of the Brauer tree of the principal $\ell$-block of $G$. This is the proposed completion of the conjecture describing the Brauer tree, determining its planar embedding in cases where that embedding is not explicitly known.