The definable finite-by-abelian-by-finite conjecture for omega-categorical pseudofinite groups

Let $G$ be a countably infinite group that is both omega-categorical, meaning that its theory has a unique countable model up to isomorphism, and pseudofinite, meaning that it is infinite and every first-order sentence true in $G$ has a finite model. A group is definably finite-by-abelian-by-finite if it has a definable normal subgroup of finite index whose derived subgroup is finite. The definable finite-by-abelian-by-finite conjecture. Every omega-categorical pseudofinite group $G$ is definably finite-by-abelian-by-finite; that is, it has a definable normal subgroup $N$ of finite index which has finite derived subgroup. This conjecture proposes a strong structural restriction on groups satisfying both finitary conditions; the supplied text does not state a resolution.