The Sylow subgroup characterization of the FSZ property

Let $G$ be a finite group. A group is called FSZ when it has the FSZ property, and a Sylow subgroup of $G$ is FSZ when it has that property.

FSZ Sylow conjecture. $G$ is FSZ if and only if all of its Sylow subgroups are FSZ.

This conjectural relation is motivated by results showing that non-FSZ properties pass from suitable normal subgroups to the full group and to Sylow normalizers, together with examples of sporadic simple groups. Whether the condition on all Sylow subgroups is sufficient in general is left open here.