  Eaton and Moret\'o proposed an extension of Brauer's famous height zero
conjecture on blocks of finite groups to the case of non-abelian defect groups,
which predicts the smallest non-zero height in such blocks in terms of local
data. We show that their conjecture holds for principal blocks of quasi-simple
groups, for all blocks of finite reductive groups in their defining
characteristic, as well as for all covering groups of symmetric and alternating
groups. For the proof, we determine the minimal non-trivial character degrees
of Sylow $p$-subgroups of finite reductive groups in characteristic~$p$. We
provide some further evidence for blocks of groups of Lie type considered in
cross characteristic.

Representation Theory;Group Theory