Brauer's height zero conjecture

Statement

Let $G$ be a finite group and let $p$ be a prime. Let $B$ be a $p$-block of $G$, that is, a block of the group algebra of $G$ over an algebraically closed field of characteristic $p$. Let $D$ be a defect group of $B$. Write $\abs{G}_p = p^a$ and $\abs{D} = p^d$. Each complex irreducible character $\chi$ in $B$ has degree $\chi(1)$ with $p$-part $\chi(1)_p = p^{a - d + h}$ for an integer $h \ge 0$, the height of $\chi$. Then every irreducible character in $B$ has height $0$ if and only if $D$ is abelian.