Ring-specific Oort conjecture for cyclic covers

Let $k$ be an algebraically closed field of characteristic $p>0$, let $f:C\rightarrow C'$ be a cyclic cover of smooth projective $k$-curves of order $p^e n$, where $(p,n)=1$, and let $W(k)$ denote the Witt vectors of $k$. Set

$$R=W(k)[\zeta_{p^e}].$$

Ring-specific Oort conjecture. The cover $f$ lifts over $R$. This is a stronger, ring-specific form of Oort's conjecture because it specifies the mixed-characteristic ring over which the lifting should occur; the source presents it as an arithmetic strengthening, and its status is not resolved in the supplied text.