Commutator-verbal implies divisibility-closed in nilpotent group

Statement

Suppose $G$ is a nilpotent group and $H$ is a commutator-verbal subgroup of $G$. Then, $H$ is a divisibility-closed subgroup of $G$. Explicitly, for any prime number $p$ such that $G$ is a $p$-divisible, $H$ is also $p$-divisible.