Existence of $L^2$ torsion for groups with finite odd-dimensional classifying spaces

Let $\Gamma$ be a discrete group whose classifying space $B\Gamma$ is a finite CW complex of odd dimension, and let $E\Gamma$ be its universal cover. Let $\phi_{E\Gamma}$ denote the $L^2$ torsion of $E\Gamma$, and let $\det(\overline H^\bullet_{(2)}(\Gamma))$ be the determinant line of the reduced $L^2$ cohomology of $\Gamma$. Existence of $L^2$ torsion for groups with finite odd-dimensional classifying spaces. One can define the $L^2$ torsion of $\Gamma$ by

$$\phi_\Gamma=\phi_{E\Gamma}\in\det(\overline H^\bullet_{(2)}(\Gamma)).$$

For residually finite or amenable groups this construction is established in the paper. The conjecture extends the definition to arbitrary discrete groups with finite odd-dimensional classifying space; the source gives no resolution in general.