Lück's twisted conjecture for von Neumann Sylvester matrix rank

Let $G$ be a group and let $\sigma:G\to\operatorname{GL}_k(\mathbb{C})$ be a homomorphism. Define $\widetilde{\sigma}:\mathbb{C}[G]\to\operatorname{Mat}_k(\mathbb{C}[G])$ by sending each $g\in G$ to $\sigma(g)g$ and extending linearly; apply it entrywise to matrices over $\mathbb{C}[G]$. Lück's twisted conjecture. For every matrix $A\in\operatorname{Mat}_{n\times m}(\mathbb{C}[G])$,

$$\operatorname{rk}_G(\widetilde{\sigma}(A))=k\cdot\operatorname{rk}_G(A).$$

The conjecture is a rank-theoretic formulation of Lück's question about twisted $L^2$-Betti numbers; the paper's abstract says that it is confirmed for sofic groups, while the general case remains open.