Uniform quasiconformality conjecture for CaTherine wheels from expanding Thurston maps

Let $f:S^1\to\mathbb{CP}^1$ be a CaTherine wheel arising from an expanding Thurston map. Uniform quasiconformality conjecture. There exists a finite constant $K$ such that $f$ is a $K$-CaTherine wheel. This is motivated by the Lattès example, where the relevant lifted Hubbard arcs are quasiarcs; the conjecture asks for analogous bounded geometric distortion for every CaTherine wheel arising from an expanding Thurston map.