On the smoothness of the conjugacy between circle maps with a break

For any $\alpha\in(0,1)$, $c\in\mathbb R_+\setminus\{1\}$ and $\gamma>0$ and for Lebesgue almost all irrational $\rho\in(0,1)$, any two $C^{2+\alpha}$-smooth circle diffeomorphisms with a break, with the same rotation number $\rho$ and the same size of the breaks $c$, are conjugate to each other via a $C^1$-smooth conjugacy whose derivative is uniformly continuous with modulus of continuity $\omega(x)=A|{\log x}|^{-\gamma}$ for some $A>0$.