The Lück approximation conjecture in the space of marked groups

Let $F$ be a finitely generated free group, let $M_k,M$ be normal subgroups of $F$, and identify $\operatorname{MG}(F)$ with the space of marked groups. Suppose that $M_k$ converges to $M$ in $\operatorname{MG}(F)$ and that $G\cong F/M$. For a matrix $A\in\operatorname{Mat}_{n\times m}(K[F])$, where $K$ is a subfield of $\mathbb{C}$, write $\operatorname{rk}_{F/M_k}(A)$ for the von Neumann rank after passing to $F/M_k$. The Lück approximation conjecture in the space of marked groups. For every such matrix,

$$\lim_{k\to\infty}\operatorname{rk}_{F/M_k}(A)=\operatorname{rk}_{F/M}(A).$$

This conjecture concerns continuity of von Neumann rank under convergence of marked groups. The paper proves the conjecture for sofic groups and consequently for one-relator groups; no general resolution is supplied here.