Solution to the Uniformly Fully Inert Subgroups Problem for Abelian Groups

A famous conjecture attributed to Dardano-Dikranjan-Rinauro-Salce states that any uniformly fully inert subgroup of a given group is commensurable with a fully invariant subgroup (see, respectively, [5] and [6]). In this short note, we completely settle this problem in the affirmative for an arbitrary Abelian group.