Dubois–Friedl–Lück conjecture on higher-order torsion and knot genus

Let $K$ be a knot in $S^3$, let $X_K$ be its exterior, and let $\phi\in H^1(X_K;\mathbb{Z})$ be a generator. A pair $(\rho,\phi)$ is assumed to be admissible, and $\deg_\phi\tau_\rho(X_K)$ denotes the degree of the corresponding higher-order Reidemeister torsion. Dubois–Friedl–Lück conjecture. There exists an epimorphism

$$\rho\colon\pi_1X_K\twoheadrightarrow\Gamma$$

onto a torsion-free elementary-amenable group such that $(\rho,\phi)$ is admissible and

$$\deg_\phi\tau_\rho(X_K)=2g(K)-1.$$

This conjecture asks whether suitable higher-order Reidemeister torsion detects the genus of every knot; the supplied text gives no resolution status.