Lück's homotopy-invariance conjecture for L2-Reidemeister torsion

Let $X$ be a finite $CW$-complex with fundamental group $\pi$. If $X$ is $L^2$-acyclic and of determinant class, define its additive $L^2$-Reidemeister torsion by

$$T^{(2)}(X):=\sum_p(-1)^p p\ln\operatorname{det}_\pi\Delta_p.$$

Lück's homotopy-invariance conjecture. $L^2$-Reidemeister torsion is a homotopy invariant.

The text says that this torsion is already an invariant of simple homotopy type and records Lück's conjecture that it is invariant under ordinary homotopy equivalence. The supplied text gives no resolution status.