Homotopy invariance of L2-torsion

Let $G$ be a group, let $\operatorname{Wh}(G)$ be its Whitehead group, and let $A\in GL_n(\mathbb ZG)$. Let $r_A^{(2)}$ be the $G$-equivariant bounded operator on $\ell^2(G)^n$ given by right multiplication by $A$, and let $\det(r_A^{(2)})$ be its Fuglede–Kadison determinant. Define

$$\Phi=\Phi^G\colon \operatorname{Wh}(G)\longrightarrow\mathbb R,$$

by $[A]\mapsto \ln(\det(r_A^{(2)}))$. Homotopy-invariance conjecture. The homomorphism $\Phi$ is trivial. This conjecture predicts that the associated $L^2$-torsion invariant vanishes on the Whitehead group. It is stated in the cited work of Lück; the source does not report a general proof or disproof.