Rigidity for circle diffeomorphisms with breaks satisfying a Zygmund smoothness condition

Let $f$ and $\tilde{f}$ be two circle diffeomorphisms with a break point, with the same irrational rotation number of bounded type, the same size of the break $c$ and satisfying a certain Zygmund type smoothness condition depending on a parameter $\gamma>2.$ We prove that under a certain condition imposed on the break size $c$, the diffeomorphisms $f$ and $\tilde{f}$ are $C^{1+\omega_{\gamma}}$-smoothly conjugate to each other, where $\omega_{\gamma}(\delta)=|\log \delta|^{-(\gamma/2-1)}.$