Normal Sylow subgroup characterization by height-zero characters

Let $G$ be a finite group and let $q$ be a prime. For each prime $p$, let $B_p(G)$ denote the principal $p$-block of $G$, and consider the irreducible characters in this block having $p$-height zero.

Normal Sylow subgroup conjecture. $G$ has a normal Sylow $q$-subgroup if and only if, for any $p\neq q$, $q$ does not divide the degree of any $p$-height zero irreducible character in $B_p(G)$.

The authors state that they know no counterexamples. They prove the corresponding assertion for solvable groups, while the general case remains open.