The conjecture on zeros of characters in 3-groups of maximal class

Let $G$ be a $3$-group of order $3^n$.

The zeros conjecture. $G$ has an irreducible character that vanishes at exactly $3^n-3^{n-1}+6$ elements if and only if $G$ is a $3$-group of maximal class with a maximal abelian subgroup.

This is expected to be the $p=3$ analogue of the corresponding equality characterization for $2$-groups; its status is left open in the source.