Free $\mathbb{Q}$-groups are residually torsion-free nilpotent

We develop a method to show that some (abstract) groups can be embedded into a free pro-$p$ group. In particular, we show that a finitely generated subgroup of a free $\mathbb{Q}$-group can be embedded into a free pro-$p$ group for almost all primes $p$. This solves an old problem raised by G. Baumslag: free $\mathbb{Q}$-groups are residually torsion-free nilpotent.