  For any prime $p>3$ and $j\in\mathbb{N}$ we construct examples of
non-$FSZ_{p^j}$ groups of order $p^{p^j+2j-1}$. In the special case of $j=1$
this yields groups of order $p^{p+1}$, which is the minimum possible order for
a non-$FSZ$ $p$-group.
