The refined Oort conjecture for cyclic covers of curves

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $\phi: Z \xrightarrow{} X$ be a cyclic $\Gamma$-cover of curves over $k$, and let $\phi_1: Y \xrightarrow{} X$ be its Galois sub-cover. Suppose $\Phi_1: \mathcal{Y}_R \xrightarrow{} \mathcal{X}_R$ is a lift of $\phi_1$ to a finite extension $R/W(k)$ in characteristic zero. The refined Oort conjecture. There exists a finite extension $R'/R$ and a lift $\Phi: \mathcal{Z}_{R'} \xrightarrow{} \mathcal{X}_{R'}$ of $\phi$ over $R'$ that contains $\Phi_1 \otimes_R R': \mathcal{Y}_{R'} \xrightarrow{} \mathcal{X}_{R'}$ as a sub-cover. This refines the Oort conjecture by requiring a lift of a cyclic cover to extend a prescribed lift of its Galois sub-cover; the paper studies this lifting problem for wildly ramified covers, and the claim is presented as a conjecture.