The minimal-zero conjecture for finite p-groups

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

$$\operatorname{mz}(p^n)=\min\{\operatorname{mz}(G)\mid |G|=p^n\},$$

where $\operatorname{mz}(G)$ is the minimum number of elements of $G$ on which a non-linear irreducible character of $G$ vanishes.

Minimal-zero conjecture.

$$\operatorname{mz}(p^n)=\operatorname{mz}(G)$$

if and only if $G$ has maximal class with an abelian maximal normal subgroup.

The conjecture is motivated by the proof of the lower bound and by computer calculations, and the source leaves it open for general $p$ and $n$.