Vanishing conjecture for $L^2$-torsion of subexponentially growing automorphisms

Let $G$ be a $\det$-finite group, meaning that the relevant Fuglede–Kadison determinants are finite, and let $\Phi\colon G\to G$ be an automorphism. Assume that $G$ has subexponential growth. Vanishing conjecture for subexponential growth. Then

$$\rho^{(2)}(\Phi)=0.$$

The conjecture is motivated by vanishing results for polynomially growing automorphisms of one-ended groups hyperbolic relative to virtually polycyclic groups. Its general status is unresolved.