Talelli's reformulated conjecture on actions on sphere times Euclidean space

Let $\Gamma$ be a torsion-free group acting smoothly and properly discontinuously on $S^n\times\mathbb{R}^k$. Talelli's reformulated conjecture. If $\Gamma$ is a torsion-free group that acts smoothly and properly discontinuously on $S^n\times\mathbb{R}^k$, then

$$cd(\Gamma)\leq k.$$

This is presented as a slightly weaker reformulation suggested by the Adem–Smith theorem, which characterizes periodic cohomology for countable groups through free, properly discontinuous, smooth actions on some $S^n\times\mathbb{R}^k$. Its general status is not resolved in the source.