The conjecture separating proper actions of free and surface groups

Let $G/H$ be a homogeneous space of reductive type. Say that $G/H$ satisfies (P-free) if it admits a proper action of a non-abelian free discrete subgroup of $G$, and satisfies (P-surf) if it admits a proper action of a discrete surface subgroup of $G$ of genus at least $2$. The free-versus-surface conjecture. The condition (P-free) does not imply (P-surf): there exists such a homogeneous space $G/H$ with a proper action of a non-abelian free discrete subgroup of $G$ but without a proper action of a discrete surface subgroup of genus at least $2$. The authors state that proving this requires a new necessary condition for (P-surf), so the conjecture remains open.