Geometric rigidity conjecture for smooth critical circle maps

A critical circle map is a $C^3$ orientation-preserving circle map with non-flat critical points; its rotation number is the associated irrational rotation invariant, and an irrational rotation number is of bounded type if there is a constant $C>0$ such that

$$\left|\theta-\frac{p}{q}\right|\geq\frac{C}{q^2}$$

for all integers $p$ and nonzero integers $q$. The odd criticality is the common odd order of the critical points.

Geometric rigidity conjecture. Any two $C^3$ critical circle maps with the same irrational rotation number of bounded type and the same odd criticality are conjugate to each other by a $C^{1+\alpha}$ circle diffeomorphism, for some $\alpha>0$.

The conjecture formalizes numerical and theoretical expectations that smooth critical circle maps with bounded-type rotation number are geometrically rigid. It has been proved for real-analytic critical circle maps by de Faria and de Melo, and later work established the result without assuming bounded type on the irrational rotation number.