Topological simplicity conjecture for acylindrically hyperbolic groups

Let $G$ be an acylindrically hyperbolic group with no non-trivial finite normal subgroups. A group topology on $G$ is topologically simple if its only closed normal subgroups are the trivial subgroup and $G$ itself. Topological simplicity conjecture. The group $G$ admits a topology with respect to which it is topologically simple. The preceding lemma shows that every acylindrically hyperbolic group admits a non-discrete Hausdorff topology, but that construction has nontrivial closed normal subgroups. The conjecture asks whether, under the stated finite-normal-subgroup hypothesis, a topology making the group topologically simple always exists.