The high-real-rank conjecture for proper surface-group actions

Let $G$ be a linear real simple Lie group and let $H$ be a reductive subgroup. Write (P-surf) for the existence of a proper action of a discrete surface subgroup of genus at least $2$, and (P-ss) for the existence of a proper action by a non-compact semisimple subgroup. High-real-rank conjecture. For any linear real simple Lie group $G$ with sufficiently high real rank, there exists a reductive subgroup $H$ such that $G/H$ is (P-surf) but not (P-ss). The source notes that this fails when the real rank is at most $3$ and suggests that the case of real rank $4$ is difficult; the conjecture is therefore open in its stated high-rank range.