Strict pairwise regularity conjecture for critical-circle conjugacies

Let $\gamma_1$ and $\gamma_2$ be critical invariant circles in the same universality class, and let $h_{\gamma_i}$ be their conjugacies to the reference dynamics. Write $H_{\gamma_1,\gamma_2}=h_{\gamma_1}\circ h_{\gamma_2}^{-1}$, and let $\kappa(\cdot)$ denote Hölder regularity. Strict pairwise regularity conjecture. For $i=1,2$,

$$\kappa(h_{\gamma_i})<\kappa(H_{\gamma_1,\gamma_2}),\qquad \kappa(h_{\gamma_i}^{-1})<\kappa(H_{\gamma_1,\gamma_2}).$$

The conjecture predicts cancellation of singularities for critical circles in the same universality class, improving on the general composition lower bound; it is presented as an expectation and remains unproved in the source.