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We prove that two $C^3$ critical circle maps with the same rotation number in a special set $𝔸$ are $C^{1+\alpha }$ conjugate for some $\alpha >0$ provided their successive renormalizations converge together at an exponential rate in the $C^0$ sense. The set $𝔸$ has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of $C^{\infty }$ critical circle maps with the same rotation number that are not $C^{1+\beta }$ conjugate for any $\beta >0$. The class of rotation numbers for which such examples exist contains...