The cocompact hyperbolic-lattice proper-action conjecture

For a homogeneous space $G/H$ of reductive type, write (P-cocH$^n$) if there exists a discrete subgroup of $G$ isomorphic to a cocompact lattice of $\operatorname{O}(n,1)$ acting properly on $G/H$, and write (P-ss) if some non-compact semisimple subgroup of $G$ acts properly on $G/H$. Cocompact hyperbolic-lattice conjecture. For every $n\geqslant 2$, there exists a homogeneous space $G/H$ of reductive type that is (P-cocH$^n$) but not (P-ss). This is presented as a stronger open question than the corresponding statement for surface groups.