Containment of the amenable radical in random-walk stabilizers

Let $G$ be a finitely generated group, and let $\boldsymbol{A_\mu}$ be the subgroup associated with a symmetric, aperiodic probability measure $\mu$ on $G$ whose support generates $G$. The amenable radical is the largest amenable normal subgroup of $G$. Amenable-radical containment conjecture. For every such measure $\mu$, the subgroup $A_\mu$ contains the amenable radical of $G$. This would establish that the subgroup detected by the random walk always contains the canonical amenable normal part of the group; the supplied text does not state whether the conjecture is known or remains open.