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 $n\geq 2$.
Comment: 22 pages, 1 table. Proofs of Theorem 3.1, Theorem 4.5 c), and Proposition 5.1 were expanded for the sake of clarity. Title was changed