Let $G_p$ be pseudovariety of all finite $p$-group. The pro-$p$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $G_p$ is a fundamental system of neighborhoods of the identity.

Margolis, Sapir and Weil showed that if a finitely generated subgroup $H$ of the free group is $G_p$-dense then $H$ contains a subgroup of rank $n$(=rank of free group) which is $G_p$-dense. In the proof of theorem 6.1, in "Dynamics of implicit operations and tameness of pseudovarieties ", Jorge used this fact in every extension closed pseudovarieties. I can not proof this and I think this not true. I appreciate any proof or counter example

Let $G_p$ be pseudovariety of all finite $p$-group. The pro-$p$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $G_p$ is a fundamental system of neighborhoods of the identity.

Margolis, Sapir and Weil showed that if a finitely generated subgroup $H$ of the free group is $G_p$-dense then $H$ contains a subgroup of rank $n$(=rank of free group) which is $G_p$-dense. In the proof of theorem 6.1, in "Dynamics of implicit operations and tameness of pseudovarieties ", Jorge Almeida used this fact in every extension closed pseudovarieties. I can not proof this and I think this not true. I appreciate any proof or counter example