Hyperbolicity of the renormalisation operator on the attractor

Let $\mathcal{R}$ be the renormalisation operator, let $\mathcal{E}_{\mathbf{c}}$ be the space of circle diffeomorphisms with fixed break data $\mathbf{c}$, and let its attractor mean the invariant attracting set of $\mathcal{R}$ in this restricted space.

Renormalisation hyperbolicity conjecture. The renormalisation operator $\mathcal{R}$, restricted to $\mathcal{E}_{\mathbf{c}}$, is hyperbolic when restricted to its attractor.

In the one-break case, hyperbolicity of the renormalisation operator is described as sufficient to identify rotation-number level sets with stable spaces and to prove the preceding rigidity conjecture. The corresponding assertion for the restricted space $\mathcal{E}_{\mathbf{c}}$ remains open.