Higher regularity conjecture for quasisymmetric maps between relative Schottky sets

Let $S$ and $\tilde S$ be relative Schottky sets of measure zero, not necessarily contained in Jordan domains, and let $f:S\to\tilde S$ be an orientation-preserving quasisymmetric map. A derivative at $p\in S$ is understood in the sense of

$$f'(p)=\lim_{q\to p,\,q\in S}\frac{f(q)-f(p)}{q-p}.$$

Higher regularity conjecture. The map $f$ is conformal at every point $p\in S$ and belongs to $C^{\infty}(S)$; that is, derivatives of all orders exist on $S$ in the sense of the displayed limit.

The conjecture predicts smooth regularity beyond the established conformality and local bi-Lipschitz continuity for quasisymmetric maps between relative Schottky sets of measure zero in Jordan domains. It is motivated by the cited work of Heinonen and Sullivan and is stated here for relative Schottky sets without the Jordan-domain assumption.