Finiteness conjecture for hyperbolic spaces of irreducible higher-rank lattices

Let $G$ be a higher-rank Lie group, or a nontrivial product of locally compact groups, and let $\Gamma$ be an irreducible lattice in $G$. A Gromov hyperbolic $G$-space is a Gromov hyperbolic metric space equipped with an isometric action of $G$; two such spaces are considered equivalent when they are coarsely equivalent. An invariant horoball is a horoball preserved by the action.

Finiteness conjecture. If $\Gamma$ is an irreducible lattice in a higher rank Lie group (or in a nontrivial product of locally compact groups) $G$, there are finitely many Gromov hyperbolic $G$-spaces, up to coarse equivalence, without invariant horoballs.

The conjecture proposes that the non-elementary hyperbolic actions of irreducible higher-rank lattices have only finitely many coarse-equivalence classes once actions with invariant horoballs are excluded. The supplied source does not indicate whether this finiteness statement has been proved or disproved.