Kleiner's CLP-to-Loewner conjecture for hyperbolic group boundaries

Let $Z$ be a compact metric space that is quasi-Möbius homeomorphic to the boundary of a hyperbolic group, and suppose that $Z$ satisfies the combinatorial Loewner property (CLP). Kleiner's conjecture. Then $Z$ is quasi-Möbius homeomorphic to a Loewner space. This conjecture asks whether the combinatorial Loewner property characterizes, up to quasi-Möbius equivalence, the classical Loewner geometry among spaces arising as hyperbolic group boundaries; the source gives no resolution.