Bounded-type regularity conjecture for Siegel-disk boundaries

Let $b1$ be a rotation number of bounded type, let $r_{b1}$ be the radius of convergence of the Siegel-disk parametrization, and define

$$\Gamma_{b1}(\theta)=\Gamma(r_{b1}e^{i\theta}).$$

Bounded-type regularity conjecture. If $b1$ is of bounded type, then the map $\Gamma_{b1}$ is at least $C^1$.

This conjecture would imply that the boundary parametrization is an embedding and hence a conjugacy for the boundary dynamics. The source provides no resolution status.