Deforming cyclic covers in towers

Obus, Wewers, and Pop recently resolved a long-standing conjecture of Oort, which asserts that every cyclic cover of a curve in characteristic p lifts to characteristic zero.Sadi subsequently asked whether these covers are also "liftable in towers."We prove that the answer to the equal-characteristic version of this question is affirmative.Our proof uses the Hurwitz tree technique along with tools developed by Obus and Wewers.