Let $\Gamma$ be a finitely generated dense subgroup of a pro-$p$ group $G$. Let $\mathbb Z_p$ be the ring of $p$-adic numbers. Denote by $\mathbb Z_p[[G]]$ the completed group algebra.

Is it true that the tensor product $\mathbb Z_p[[G]]\otimes_{\mathbb Z_p[\Gamma]} \mathbb Z_p[[G]]$ is torsion-free as an abelian group?