The universal $L^2$-torsion characterization of free-group isomorphisms

Let $F_1$ and $F_2$ be finitely generated free groups, let $\varphi\colon F_1\to F_2$ be a homomorphism, and let $\tau^{(2)}_u(\varphi)$ denote its universal $L^2$-torsion. Universal $L^2$-torsion conjecture. The homomorphism $\varphi$ is an isomorphism if and only if

$$\tau^{(2)}_u(\varphi)=1.$$

The conjecture seeks to determine how much group-theoretic information universal $L^2$-torsion detects. The source notes that Jaikin-Zapirain proved an $L^2$-homology characterization in the equal-rank injective case via compressed subgroups, while the stated equivalence for arbitrary homomorphisms between finitely generated free groups remains unresolved.