Universality conjecture for the regularity of critical invariant circles

Let $R$ denote a critical invariant circle, and let $\kappa(R)$ denote its regularity. A numerical characteristic is universal when it takes the same value on an open set of functions. Critical-circle regularity universality conjecture. The regularity $\kappa(R)$ of the critical invariant circle is a universal number. The claim concerns the expected independence of this regularity from details of the map, as suggested by renormalization-group convergence; the source gives no proof and refers to more precise formulations elsewhere.