Rigidity conjecture for multicritical circle maps with the same signature

Let $f$ and $g$ be $C^3$ multicritical circle maps without periodic points. For a multicritical circle map $f$ with critical points $c_i=c_i(f)$, $0\leq i\leq n_f-1$, irrational rotation number $\rho(f)$, unique invariant Borel probability measure $\mu_f$, critical exponents $s_i$, and $\lambda_i=\mu_f[c_i,c_{i+1})$ with $c_{n_f}=c_0$, its signature is the tuple

$$(\rho(f),n_f;s_0,s_1,\ldots,s_{n_f-1};\lambda_0,\lambda_1,\ldots,\lambda_{n_f-1}).$$

Rigidity conjecture. If $f$ and $g$ have the same signature, then they are conjugate by a $C^1$ diffeomorphism. Moreover, if their common rotation number is of bounded type, then the conjugacy is $C^{1+\alpha}$ for some universal $\alpha>0$. The conjecture seeks differentiable rigidity beyond the quasisymmetric rigidity theorem, which requires the conjugacy to map corresponding critical points but does not require equal critical exponents. The $C^1$ statement was conjectured as sufficient for differentiability; the stronger real-analytic case has been proved through work of de Faria, de Melo, Yampolsky, and Khanin–Teplinsky, so the conjecture is solved in that case, while the stated $C^3$ formulation is not established by the supplied context.