  Let $F$ be a free group of arbitrary rank and let $H$ be a finitely generated
subgroup of $F$. Given a pseudovariety $\mathbf{V}$ of finite groups, i.e. a
class of finite groups closed under taking subgroups, quotients and finitary
direct products, we endow $F$ with its pro-$\mathbf{V}$ topology. Our main
result states that it is decidable whether $H$ is $\mathbf{Su}$-dense, where
$\mathbf{Su}\subset \mathbf{S}$ denote respectively the pseudovarieties of all
finite supersolvable groups and all finite solvable groups. Our motivation
stems from the following open problem: is it decidable whether $H$ is
$\mathbf{S}$-dense?
