The 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 $\operatorname{actdim}(G)$ be the least dimension of a contractible manifold admitting a proper $G$-action. Action dimension conjecture.

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

This conjecture would imply the action-dimension obstruction for products of non-abelian free groups and is one of the vanishing conjectures relating $L^2$-homology to dimensions of group actions. The source presents it as an open conjecture.