Weak equivalence to Bernoulli shifts for some algebraic actions

Namely, we prove that if $G$ is a countable, discrete group and $f\in M_{n}(\Z(G))$ is invertible on $\ell^{2}(G)^{\oplus n},$ but $f$ is not invertible in $M_{n}(\Z(G))$, then the measure-preserving action of $G$ on $X_{f}$ equipped with the Haar measure is weakly equivalent to a Bernoulli action. We shall in fact prove this weak equivalence in the case that $f$ has a "formal inverse in $\ell^{2}$".