Let $G$ be a torsion-free group and assume that the integral group ring $\mathbb{Z}G$ is torsion-free as well. Let $M$ be a torsion-free, finitely generated module over $\mathbb{Z}G$.

If we assume that $M \otimes_{\mathbb{Z}G} \mathbb{Q}G$ is a projective $\mathbb{Q}G$-module, can we conclude that $M$ itself is projective?

(For finite groups $G$ a somewhat similar question was already asked here: Looking for criterion for $\mathbb{Z}G$-modules to be projective.)

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