Commutator-verbal implies divisibility-closed in nilpotent group

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a nilpotent group. That is, it states that in a Nilpotent group (?), every subgroup satisfying the first subgroup property (i.e., Commutator-verbal subgroup (?)) must also satisfy the second subgroup property (i.e., Divisibility-closed subgroup (?)). In other words, every commutator-verbal subgroup of nilpotent group is a divisibility-closed subgroup of nilpotent group.
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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.

In particular, this also shows that $H$ is a powering-invariant subgroup of $G$.