Commensurability and QI classification of free products of finitely generated abelian groups

Suppose a group $G$ is quasi-isometric to a free product of a finite set $S$ of finitely generated abelian groups; let $S^{\mathrm{\prime }}$ denote the set of ranks of the free abelian parts of the groups in $S$. Then $G$ is commensurable with the free product of $\mathbb{Z}$ with a ${\mathbb{Z}}^n$ for each $n$ occurring in $S^{\mathrm{\prime }}$.