Lower central series members need not be completely divisibility-closed in nilpotent group

Statement
It is possible to have a nilpotent group $$G$$ such that the nontrivial members of the lower central series of $$G$$ are not completely divisibility-closed subgroups of $$G$$.

Related facts

 * Upper central series members are completely divisibility-closed in nilpotent group
 * Lower central series members are divisibility-closed in nilpotent group

Proof
For a class $$c$$ example, let $$G = UT(c + 1,\mathbb{Q}) \times \mathbb{Q}/\mathbb{Z}$$. $$G$$ is divisible by all primes. However, all its lower central series members, which are completely contained in the first direct factor, are not completely divisibility-closed in $$G$$ because they miss all the torsion of the second direct factor.