Anosov groups that are indiscrete in rank one

We exhibit Anosov subgroups of ${\mathsf{S}\mathsf{L}}_d(\mathbb{R})$ that do not embed discretely in any rank-$1$ simple Lie group of noncompact type, or indeed, in any finite product of such Lie groups. These subgroups are isomorphic to free products $\mathrm{\Gamma }\ast \mathrm{\Delta }$, where $\mathrm{\Gamma }$ is a uniform lattice in ${\mathsf{F}}_4^{(-20)}$ and $\mathrm{\Delta }$ is a uniform lattice in $\mathsf{S}\mathsf{p}(m,1)$, $m\ge 51$.