Friedl–Lück–Tillmann's amenable-group $L^2$-torsion polytope conjecture

Let $G$ be an $L^2$-acyclic group of type $\mathtt{F}_{}$ satisfying the Atiyah conjecture, with $\operatorname{Wh}(G)=0$. Let $P_{L^2}(G)$ denote its $L^2$-torsion polytope.

Friedl–Lück–Tillmann's conjecture. If $G$ is amenable but not virtually $\mathbb{Z}$, then

$$P_{L^2}(G)\text{ is a singleton}.$$

The paper states that its theorem confirms this conjecture. Thus, with the hypotheses above, the assertion is established in the source.