Kleiner–Bourdon–Kleiner conjecture on the Combinatorial Loewner Property

Let $Z$ be an approximately self-similar metric space. Suppose that $Z$ satisfies the Combinatorial Loewner Property, a quasisymmetric-invariant property defined using discrete approximations of $Z$.

Kleiner–Bourdon–Kleiner conjecture. If $Z$ satisfies the Combinatorial Loewner Property, then $Z$ is quasisymmetric to an Ahlfors regular Loewner space.

This conjecture asks whether the Combinatorial Loewner Property is sufficient to produce a Loewner metric in the quasisymmetry class of an approximately self-similar space. It remains open even for the classical Sierpiński carpet, although there are interesting examples of planar carpets for which the conclusion holds.