Davis–Okun conjecture on action dimension and $L^2$-Betti numbers

Let $G$ be a group, and let its $i$th $L^2$-Betti number be the corresponding $L^2$-homological invariant. The action dimension $\operatorname{actdim} G$ is the least dimension of a contractible manifold on which $G$ acts properly and cocompactly.

Davis–Okun conjecture. If the $i$th $L^2$-Betti number of $G$ is nonzero, then

$$\operatorname{actdim} G\geq 2i.$$

The conjecture is proposed as a connection between action dimension and $L^2$-cohomology; the computations in the paper provide evidence for it, but no resolution is supplied here.