Moore's conjecture for strongly graded rings

Let $R$ be an arbitrary ring with unit element $1$, let $\Gamma$ be a group, and let $R(\Gamma)$ be a strongly graded ring. Let $H$ be a subgroup of finite index in $\Gamma$, and let $M$ be an $R(\Gamma)$-module.

Moore's conjecture. Assume that for every nontrivial element $x$ in $\Gamma$, at least one of the following conditions holds: $M1)$ $\langle x\rangle\cap H\neq\\{e\\}$; or $M2)$ $\operatorname{ord}(x)$ is finite and invertible in $R$. Then every $R(\Gamma)$-module $M$ that is projective over $R(H)$ is also projective over $R(\Gamma)$.

This conjecture generalizes the original formulation for group rings and is also formulated for crossed products and strongly graded rings. The source states that it is known for various classes of groups, while the general conjecture remains open; its profinite analogue is proved for arbitrary profinite groups.