In 1974 Haim Gaifman conjectured that if a first-order theory T is relatively categorical over T(P) (the theory of the elements satisfying P), then every model of T(P) expands to one of T.
The conjecture has long been known to be true in some special cases, but nothing general is known. I prove it in the case of abelian groups with distinguished subgroups. This is some way outside the previously known cases, but the proof depends so heavily on the Kaplansky-Mackey proof of Ulm's theorem that the jury is out on its generality.