The local-freeness conjecture for homological dimension one

For a group $G$, write $\operatorname{hd}_{\mathbb Z}(G)$ for its homological dimension over $\mathbb Z$. A group is locally free if every finitely generated subgroup is free.

Local-freeness conjecture. Every group $G$ satisfying

$$\operatorname{hd}_{\mathbb Z}(G)\leq 1$$

is locally free.

This is the $k=\mathbb Z$ specialization of the filtered-colimit conjecture for homological dimension one. The converse direction is open even over $\mathbb Z$; the source notes that a counterexample would also give a counterexample to the Atiyah conjecture on integrality of finite von Neumann dimensions for torsion-free groups.