Moore's conjecture on projectivity over finite-index subgroup extensions

Let $G$ be a group, let $H$ be a subgroup of finite index, and let $R$ be an arbitrary ring. Assume that for every nontrivial element $x$ of $G$, at least one of the following conditions holds: $\text{M1)}$ $\langle x\rangle\cap H\neq\\{e\\}$, in particular if $G$ is torsion free; or $\text{M2)}$ $\operatorname{ord}(x)$ is finite and invertible in $R$. Moore's conjecture. Every $RG$-module $M$ that is projective over $RH$ is also projective over $RG$. Moore's conjecture is an infinite analogue of Chouinard's theorem and implies Serre's result that a torsion-free group with a finite-index subgroup of finite cohomological dimension has the same finite cohomological dimension. The conjecture is stated here without evidence of resolution.