On minimal positive heights for blocks of almost quasi-simple groups

The Eaton–Moretó conjecture extends the recently-proven Brauer height zero conjecture to blocks with non-abelian defect group, positing equality between the minimal positive heights of a block of a finite group and its defect group. Here we provide further evidence for the inequality in this conjecture that is not implied by Dade’s conjecture. Specifically, we consider minimal counter-examples and show that these cannot be found among almost quasi-simple groups for $p\ge 5$. Along the way, we observe that most such blocks have minimal positive height equal to~1.