  Let $G=\prod_{i=1}^{r} G_i$ be a product of simple real algebraic groups of
rank one and $\Gamma$ an Anosov subgroup of $G$ with respect to a minimal
parabolic subgroup. For each $v$ in the interior of a positive Weyl chamber,
let $\mathcal R_v\subset\Gamma\backslash G$ denote the Borel subset of all
points with recurrent $\exp (\mathbb R_+ v)$-orbits. For a maximal
horospherical subgroup $N$ of $G$, we show that the $N$-action on ${\mathcal
R}_v$ is uniquely ergodic if $r={rank}(G)\le 3$ and $v$ belongs to the interior
of the limit cone of $\Gamma$, and that there exists no $N$-invariant {Radon}
measure on $\mathcal R_v$ otherwise.
