  We characterize when a finite group G possesses a Sylow 3-subgroup P with
abelianization of order 9 in terms of the number of height zero characters
lying in the principal 3-block of G, settling a conjecture put forward by
Navarro, Sambale, and Tiep in 2018. Along the way, we show that a recent result
by Laradji on the number of character of height zero in a block that lie above
a given character of some normal subgroup holds, without any hypothesis on the
group for blocks of maximal defect.
