Talelli's conjecture on torsion-free groups with periodic cohomology

Let $\Gamma$ be a group. Its cohomological dimension is

$$cd(\Gamma)=\sup\{n\mid H^n(\Gamma;M)\neq 0\text{ for some }\mathbb{Z}\Gamma\text{-module }M\}.$$

The group $\Gamma$ has periodic cohomology after $k$ steps if there exists an integer $q>0$ such that $H^i(\Gamma,-)$ and $H^{i+q}(\Gamma,-)$ are naturally isomorphic functors for all $i>k$. Talelli's conjecture. A torsion-free group $\Gamma$ that has periodic cohomology after some steps has finite cohomological dimension. The conjecture is known for the class of $LH\mathcal{F}$-groups, which includes all linear and all elementary amenable groups; its validity in general remains open.