The cocompact action dimension conjecture for $L^2$-homology

For a discrete group $G$, let $L^2H_i(G)$ denote the $L^2$-homology of a contractible $G$-space, and let $\operatorname{cadim}(G)$ be the least dimension of a contractible $G$-manifold. Cocompact action dimension conjecture.

$$L^2H_i(G)=0\quad\text{for }i>\operatorname{cadim}(G)/2.$$

The source notes that no group is known there with $\operatorname{actdim}(G)<\operatorname{cadim}(G)$, and that this conjecture implies the Singer conjecture. Its general validity remains open.