Quasisymmetric uniformization target for random carpets

Determine a canonical quasisymmetric uniformizing space for random metric carpets arising from stochastic models, in particular for the conformal loop ensemble CLE_κ carpets with κ in (8/3, 4], which are almost surely homeomorphic to the standard Sierpiński carpet but are not quasisymmetrically equivalent to any round carpet. Identify an appropriate canonical target (or class of targets) that plays the role analogous to round carpets in the deterministic setting.

Background

The paper proves that for κ in (8/3, 4], the CLE_κ space is almost surely a metric carpet (homeomorphic to the standard Sierpiński carpet) but not a quasiround carpet. In classical (deterministic) settings, round carpets serve as canonical quasisymmetric uniformization targets under suitable geometric hypotheses.

Given that CLE_κ carpets fail to be quasiround, the authors ask what the correct canonical uniformization target should be in the stochastic setting, aiming to parallel the role of round carpets in geometric group theory and complex dynamics.