Approximation conjecture for $L^2$-torsion

Let $X$ be a finite connected $CW$-complex and let $\overline{X} \to X$ be a $G$-covering. Approximation conjecture for $L^2$-torsion. If the $G$-$CW$-structure on $\overline{X}$ and, for each $i \in I$, the $CW$-structure on $G_i\backslash \overline{X}$ come from a given $CW$-structure on $X$, then

$$\rho^{(2)}(\overline{X};{\mathcal N}(G)) = \lim_{i \to \infty} \frac{\rho^{(2)}(G_i\backslash \overline{X};{\mathcal N}(\{1\}))}{[G:G_i]}.$$

If $X$ is a closed Riemannian manifold and the quotient spaces and $\overline{X}$ carry the induced Riemannian metrics, the torsion in this equality should also be replaceable by the analytic versions. If $b_n^{(2)}(\overline{X};{\mathcal N}(G))$ vanishes for all $n \ge 0$, then

$$\rho^{(2)}(\overline{X};{\mathcal N}(G)) = \lim_{i \to \infty} \frac{\rho^{{\mathbb Z}}(G_i\backslash \overline{X})}{[G:G_i]}.$$

These conjectures concern approximation of $L^2$-torsion and related invariants by finite quotients; the paper proves them in a special case, while the general assertions remain open.