Universality conjecture for regularity between pairs of critical circles

Let $\gamma_1$ and $\gamma_2$ be critical invariant circles, and let $G_{\gamma_1,\gamma_2}$ and $H_{\gamma_1,\gamma_2}$ denote the associated maps between them. Let $\kappa(\cdot)$ denote regularity, and call a numerical characteristic universal when it takes the same value on an open set of functions. Pairwise critical-circle regularity universality conjecture. For pairs of critical circles $\gamma_1$ and $\gamma_2$, the regularities $\kappa(G_{\gamma_1,\gamma_2})$ and $\kappa(H_{\gamma_1,\gamma_2})$ are universal numbers. The claim predicts that these inter-circle regularities are independent of the details of the map within a universality domain; the source does not provide a proof.