Weak marginality is direct power-closed

Statement
Suppose $$H$$ is a weakly marginal subgroup of a group $$G$$, and $$\alpha$$ is a finite or infinite cardinal. Then, in the direct power $$G^\alpha$$, $$H^\alpha$$ is a weakly marginal subgroup. In fact, it is weakly marginal for the same collection of word-letter pairs for which $$H$$ is weakly marginal in $$G$$.

Related facts

 * Marginality is direct power-closed