Conjecture on jump cohomology and finite cohomological dimension for groups without R-torsion

Let $G$) be a group without $R$-torsion, where $R$ is a commutative ring with a unit, and let $k$ be a nonnegative integer. Jump cohomology conjecture. $G$ has jump cohomology of height $k$ over $R$ if and only if $G$ has finite cohomological dimension $k$ over $R$. The forward implication was conjectured in the cited work; the converse follows immediately from $cd_R(G)=k$, while the reverse direction is known when $G$ belongs to ${\sl H}{\mathcal F}$ and remains open in general.