The nilpotent-group conjecture for the sum of element orders

Let $G$ be a finite nilpotent group, and let $S(G)$ denote the sum of the orders of all elements of $G$.

Nilpotent-group conjecture. One has

$$S(G)\geq |G|,$$

and equality holds if and only if $G$ is cyclic.

This extends the preceding results for non-abelian $p$-groups with cyclic maximal subgroups. The supplied text gives no resolution of the conjecture for all finite nilpotent groups.