The combinatorial Loewner property implies quasi-Möbius equivalence to a Loewner space

Let $Z$ be a metric space that is quasi-Möbius equivalent to the boundary of a hyperbolic group. The combinatorial Loewner property (CLP) is the property referred to in the claim.

CLP-to-Loewner conjecture. If $Z$ satisfies the CLP, then $Z$ is quasi-Möbius equivalent to a Loewner space.

This conjecture would explain the significance of the CLP for boundaries of hyperbolic spaces: it would promote the combinatorial condition to the analytic Loewner property up to quasi-Möbius equivalence. Its general status remains open.