Realization of the amenable radical by a random-walk subgroup

Let $G$ be a finitely generated group. For a probability measure $\mu$ on $G$, let $A_\mu$ denote the associated subgroup, and let the amenable radical be the largest amenable normal subgroup of $G$. Amenable-radical realization conjecture. There exists some probability measure $\mu$ on $G$ such that

$$A_\mu=\text{the amenable radical of }G.$$

This is presented as a stronger or more desirable formulation concerning whether the amenable radical can be exactly recovered from a suitable random walk. The supplied text does not specify additional conditions on $\mu$ or give evidence that the conjecture has been resolved.