Bounded-type nonsmoothness conjecture for Siegel-disk boundaries

Let $b1$ be a rotation number of bounded type, and let $\Gamma_{b1}(\theta)$ denote the parametrization of the boundary of the corresponding Siegel disk. Bounded-type nonsmoothness conjecture. There exists a real number $p$ with $1<p<3$ such that $\Gamma_{b1}$ is not of class $C^p$.

The claim refines the preceding $C^1$ regularity conjecture by predicting limited smoothness of the boundary. The source reports numerical evidence and mentions an estimate around $1.7$ for the last noble invariant circle, but gives no proof.