I am looking for a reference for (or a proof of) the following fact:

Let $G$ be a profinite group. Let $X^\bullet$ be a complex of discrete $G$-modules. We assume that the cohomology $G$-modules of $X^\bullet$ are nontrivial only for finitely many degrees, and that they are finitely generated over $\mathbb{Z}$. (We do not assume that the $G$-modules $X^i$ are finitely generated over $\mathbb{Z}$.) Then there exists a quasi-isomorphism $M^\bullet \to X^\bullet$, where $M^\bullet$ is a bounded complex of finitely generated (over $\mathbb{Z}$) torsion-free $G$-modules.