Laitinen's conjecture on the primitivity invariant of finite groups

Let $G$ be a finite group, and let $\operatorname{prim}(G)$ denote its primitivity invariant and $\operatorname{Sm}(G)$ its Smith invariant. Laitinen's conjecture. If

$$\operatorname{prim}(G)\geq 2,$$

then

$$\operatorname{Sm}(G)\neq 0.$$

The conjecture relates the existence of sufficiently many primitive representations to nonvanishing Smith data. Its status is left open here because the supplied text gives no resolution evidence.