Existence of transverse subgroups that are not hypertransverse

Determine whether there exists a θ-transverse subgroup Γ < G (for some connected semisimple real algebraic group G and non-empty subset θ of simple roots) that is not θ-hypertransverse; specifically, ascertain whether one can construct a θ-transverse subgroup Γ that does not admit a proper geodesic Gromov hyperbolic space Z with a properly discontinuous isometric Γ-action and a Γ-equivariant homeomorphism from the limit set Λ^Z ⊂ ∂Z to the θ-limit set Λ_Γ^θ ⊂ F_θ.

Background

The paper introduces θ-transverse subgroups of a semisimple real algebraic group G, defined by θ-regularity and θ-antipodality, and then refines this to θ-hypertransverse subgroups that additionally admit an action on a proper geodesic Gromov hyperbolic space Z with a Γ-equivariant identification of the Gromov boundary limit set with the θ-limit set in F_θ. The authors note that many known examples (Anosov and relatively Anosov groups and their subgroups) are hypertransverse.

Despite this abundance of examples, the authors explicitly state that they lack a counterexample of a transverse subgroup that fails to be hypertransverse, raising the question of whether the two notions coincide or whether a non-hypertransverse transverse subgroup exists. This uncertainty is central to clarifying the scope of their rigidity and ergodicity results, which are proved under the hypertransverse hypothesis.