C-closed implies completely divisibility-closed in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group and $$H$$ is a c-closed subgroup of $$G$$, i.e., $$H$$ equals the centralizer $$C_G(K)$$ for some subgroup $$K$$ of $$G$$. Then, $$H$$ is a completely divisibility-closed subgroup of $$G$$: for any prime number $$p$$ such that every element of $$G$$ has a $$p^{th}$$ root in $$G$$, it is true that all $$p^{th}$$ roots of any element of $$H$$ are in $$H$$.

Facts with similar proof technique

 * Upper central series members are completely divisibility-closed in nilpotent group
 * Powering-invariance is commutator-closed in nilpotent group