The local Oort conjecture for cyclic groups

Let $k$ be an algebraically closed field of characteristic $p$, and let $G$ be a finite group. A $G$-Galois extension $k[[z]]/k[[t]]$ is an integral extension of integrally closed domains that is $G$-Galois on the level of fraction fields. The local lifting problem asks whether there exists a discrete valuation ring $R$ of characteristic zero with residue field $k$ and a $G$-Galois extension $R[[Z]]/R[[T]]$ reducing to $k[[z]]/k[[t]]$, with $Z$ and $T$ reducing to $z$ and $t$, respectively.

Local Oort conjecture. The local lifting problem can always be solved when $G$ is cyclic.

Through the local-global principle, this would imply that every Galois cover of $k$-curves with cyclic inertia groups lifts to characteristic zero, even when its global Galois group is not cyclic.