  Given a group $G$ and a family of subgroups $\mathcal{F}$, we consider its
classifying space $E_{\mathcal F}G$ with respect to $\mathcal{F}$. When
$\mathcal F = \mathcal{VC}yc$ is the family of virtually cyclic subgroups,
Juan-Pineda and Leary conjectured that a group admits a finite model for this
classifying space if and only if it is virtually cyclic. By establishing a
connection to conjugacy growth we can show that this conjecture holds for
linear groups. We investigate a similar question that was asked by
L\"uck--Reich--Rognes--Varisco for the family of cyclic subgroups. Finally, we
construct finitely generated groups that exhibit wild inner automorphims but
which admit a model for $E_{\mathcal{VC}yc}(G)$ whose 0-skeleton is finite.
