Homotopy invariance of $L^2$ torsion

Let $M$ and $N$ be compact odd-dimensional manifolds, and let $f:M\to N$ be a homotopy equivalence. Let $\widehat M$ and $\widehat N$ be their normal $\Gamma$-covering spaces, and let $\phi_{\widehat M}$ and $\phi_{\widehat N}$ denote the associated $L^2$ torsion elements in the determinant lines of reduced $L^2$ cohomology. Suppose that $\widehat M$ is of determinant class, equivalently that $\widehat N$ is of determinant class. Via the identification of determinant lines induced by $f$, Homotopy invariance of $L^2$ torsion. one has

$$\phi_{\widehat M}=\phi_{\widehat N}\in\det{\bar H}_{(2)}^\bullet(\widehat M).$$

This is the homotopy-invariance conjecture for $L^2$ torsion, originally formulated by Lück in the case of vanishing $L^2$ cohomology. The paper proves the equality when the covering transformation group is residually finite or amenable; the general statement remains open in the source.