  Motivated by a question of M. Kapovich, we show that the $\mathbb{Z}^2$
subgroups of $\mathsf{SL}_3(\mathbb{R})$ that are regular in the language of
Kapovich--Leeb--Porti, or divergent in the sense of Guichard--Wienhard, are
precisely the lattices in minimal horospherical subgroups. This rules out any
relative Anosov subgroups of $\mathsf{SL}_3(\mathbb{R})$ that are not in fact
Gromov-hyperbolic. By work of Oh, it also follows that a Zariski-dense discrete
subgroup $\Gamma$ of $\mathsf{SL}_3(\mathbb{R})$ contains a regular
$\mathbb{Z}^2$ if and only if $\Gamma$ is commensurable to a conjugate of
$\mathsf{SL}_3(\mathbb{Z})$. In particular, a Zariski-dense regular subgroup of
$\mathsf{SL}_3(\mathbb{R})$ contains no $\mathbb{Z}^2$ subgroups.
