Decorated rotation number determines smooth rigidity classes for circle diffeomorphisms with breaks

A circle diffeomorphism with breaks is a circle diffeomorphism having break points; the decorated rotation number is the rotation number together with the additional break data, and the size of a break is its associated break invariant. Let $\gamma$ range over decorated rotation numbers.

Rigidity conjecture. There is a full-measure set of decorated rotation numbers $\gamma$ such that, whenever $T_1$ and $T_2$ are circle diffeomorphisms with breaks having breaks of the same size and the same decorated rotation number, $T_1$ and $T_2$ are smoothly conjugate.

This conjecture concerns the characterization of smooth rigidity classes by renormalization data. In the one-break case, restricted to irrational rotation numbers, hyperbolicity of the renormalization operator and the description of rotation-number level sets as stable spaces are stated to imply the conjecture; the general statement remains open.