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) 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, and in the first part of this MO post claims a nilpotent and torsion-free $G$ is embedded into $\hat G$ as a lattice, but without proofs in either case.

Any comments or references are appreciated.