Eaton–Moretó conjecture on minimal positive heights of blocks

Let $p$ be a prime, let $G$ be a finite group, and let $B$ be a $p$-block of $G$ with defect group $D$. For a finite $p$-group $D$, define

$$\operatorname{mh}(D):=\min\{\infty,\log_p(\chi(1))\mid \chi\in\operatorname{Irr}(D),\,\chi(1)>1\}$$

with value $\infty$ when $D$ is abelian. Define $\operatorname{mh}(B)$ as the minimal positive height of an irreducible complex character in $B$, with value $\infty$ if no such character exists. Eaton–Moretó conjecture.

$$\operatorname{mh}(B)=\operatorname{mh}(D).$$

This extends Brauer’s height zero conjecture from abelian to arbitrary defect groups by asserting equality of the minimal positive heights of a block and its defect group. It has been verified in several cases and for various classes of groups and block types, but remains open in general, even for solvable groups.