On the uniqueness of affine IETs semi-conjugated to IETs

We prove that for almost every irreducible interval exchange transformation $T$ and for any vector $\omega$ in its associated central-stable space (with respect to the Kontsevich-Zorich cocycle) there exists a unique AIET, up to normalization of its domain, semi-conjugated to $T$ and whose log-slope vector equals $\omega$. This provides a partial answer to a question raised by S. Marmi, P. Moussa, and J.-C. Yoccoz.