The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by $$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$ the group-like part of the completed (by the augmentation ideal $I$) group ring.

Question: Suppose that $G$ is residually nilpotent and torsion-free. Then, is the natural map $G\to \hat G$ an injection?

I've heard that this is true for free groups, but without any proofs. If $G$ is further nilpotent, I could prove it using the fact that $G_n = (1+I^n)\cap G$ (for the notation and the proof, see this MO post and Theorem 12.1.6 of CDBooK by Chmutov-Duzhin-Mostovoy, respectively).

Any comments or references are appreciated.