The matrix-kernel approximation conjecture for residual group towers

Let $\pi$ be a group with a nested sequence of normal subgroups $\pi=\pi_1\supset\pi_2\supset\cdots$ and $\bigcap_i\pi_i=\{1\}$. For a matrix $A\in M(d\times d,\mathbb Z\pi)$, let $A_i$ be its image under the quotient map $p_i$ to $M(d\times d,\mathbb Z[\pi/\pi_i])$.

Matrix-kernel approximation conjecture.

$$\lim_{i\to\infty}\dim_{\pi/\pi_i}(\ker A_i)=\dim_\pi(\ker A).$$

The source states that this matrix conjecture implies, and is equivalent to, the preceding approximation conjecture for $L^2$-Betti numbers. The supplied text gives no resolution status.