On projectively inert subgroups of completely decomposable finite rank groups

Let a group $G$ be a finite direct sum of torsion-free rank $1$ groups $G_{i}$. It is proved that every projectively inert subgroup of $G$ is commensurate with a fully invariant subgroup if and only if all $G_{i}$ are not divisible by any prime number $p$, and for different subgroups $G_{i}$ and $G_{j}$ their types are either equal or incomparable.