The B+E characterization conjecture for generalized Bassian groups

Let $G$ be an abelian group. Say that $G$ is B+E if it decomposes as $G=A\oplus E$, where $A$ is Bassian and $E$ is elementary. A group is B+E characterization conjecture. B+E if and only if it is generalized Bassian.

This would establish the converse of the implication proved in the paper that every generalized Bassian group is B+E, thereby completely characterizing generalized Bassian groups within the class of mixed abelian groups with bounded $p$-torsion. The supplied text does not indicate whether the converse is resolved.