Moore's conjecture on projectivity over finite-index subgroups

Let $\Gamma$ be a group and $H$ a subgroup of finite index. Moore's condition is that no elements of prime order lie in $\Gamma-H$. Let $M$ be a $\mathbb{Z}\Gamma$-module.

Moore's conjecture. If $M$ is projective over $\mathbb{Z}H$, then it is projective over $\mathbb{Z}\Gamma$. In particular, for torsion-free $\Gamma$, projectivity over $\mathbb{Z}\Gamma$ is equivalent to projectivity over $\mathbb{Z}H$.

This conjecture generalizes Serre's theorem on cohomological dimensions of torsion-free groups and finite-index subgroups. The paper proves it for groups in Kropholler's hierarchy $\mathbf{LH}\mathscr{F}$ and establishes additional closure properties and examples, but the general conjecture remains open.