The Lück approximation conjecture

Let $\Gamma$ be a group, let $K$ be a field, and let

$$\Gamma>N_1>N_2>\ldots$$

be a descending chain of normal subgroups of finite index with trivial intersection. For a matrix $A$ over $K[\Gamma]$, define

$$\operatorname{rk}_{\Gamma/N_i}(A)=\frac{\dim_K\operatorname{Im}\phi_{\Gamma/N_i}^A}{|\Gamma:N_i|},$$

where $\phi_{\Gamma/N_i}^A:K[\Gamma/N_i]^n\to K[\Gamma/N_i]^m$ is right multiplication by $A$. The Lück approximation conjecture. The sequence $\{\operatorname{rk}_{\Gamma/N_i}(A)\}_{i\geq 1}$ converges; its limit is independent of the chain, and, if $\Gamma$ is locally indicable, there exists a universal embedding $K[\Gamma]\to\mathcal Q$ such that

$$\lim_{i\to\infty}\operatorname{rk}_{\Gamma/N_i}(A)=\operatorname{rk}_{K[\Gamma]}(A).$$

This is an approximation statement relating ranks over finite quotients to the rank over the group algebra. The supplied text does not state whether it is proved or refuted, so its status remains open.