Positive laws on word values in residually-p groups

We address the following problem: if the set of all values of a word w in a group G   satisfies a positive law, does it follow that the whole verbal subgroup w(G)w(G) also satisfies a positive law? In the realm of finitely generated residually-p groups, we obtain a positive answer for the simple commutator words for all but finitely many primes p, depending on the positive law. Furthermore, if we assume that the set of all powers of the values of w   satisfies a positive law, then the conclusion holds for all primes. We extend these results to any outer commutator word, in the case that the verbal subgroup w(G)w(G) is finitely generated.