The manifold form of the cocompact action dimension conjecture

Let $(M,\partial M)$ be an $n$-manifold with contractible components that admits a geometric group action, and let $L^2H_i(M)$ denote its $L^2$-homology. Manifold cocompact action dimension conjecture.

$$L^2H_i(M)=0\quad\text{for }i>n/2.$$

This is presented as an equivalent manifold formulation of the cocompact action dimension conjecture, because a contractible $G$-manifold can be used to compute $L^2H_i(G)$. The source leaves the general assertion open.