Classifying spaces for the family of virtually abelian subgroups of orientable $3$-manifold groups

For a group $G$, let ${\mathcal{F}}_n$ be the family of all the subgroups of $G$ containing a subgroup isomorphic to ${\mathbb{Z}}^r$ for some $r=0,1,2,\dots ,n$ of finite index. Joecken, Lafont and S'anchez Salda~na computed the ${\mathcal{F}}_1$-dimension of 3-manifold groups. The goal of this article is to compute the ${\mathcal{F}}_n$-geometric dimension of 3-manifold groups for all $n\ge 2$.