The independence conjecture for von Neumann Sylvester matrix rank

Let $G$ be a group, let $K$ be a field, and let $\phi_1,\phi_2:K\to\mathbb{C}$ be embeddings of $K$ into $\mathbb{C}$. For a matrix $A\in\operatorname{Mat}_{n\times m}(K[G])$, write $\phi_i(A)$ for the matrix obtained by applying $\phi_i$ to the coefficients of $A$. The independence conjecture. For every such matrix,

$$\operatorname{rk}_G(\phi_1(A))=\operatorname{rk}_G(\phi_2(A)).$$

This conjecture was proved for sofic groups and for locally indicable groups, while the statement for general groups remains open.