The local refined Oort conjecture for cyclic extensions

Let $k$ be the residue field of a discrete valuation ring $R$ of characteristic zero, and let $k[[z_2]]/k[[x]]$ be a cyclic $\Gamma$-extension with Galois sub-extension $k[[z_1]]/k[[x]]$. Suppose $R[[Z_2]]/R[[X]]$ is a lift of the sub-extension. The local refined Oort conjecture. There exist a finite extension $R'$ of $R$ and a $\Gamma$-Galois extension $R'[[Z_2]]/R'[[X]]$ that lifts $k[[z_2]]/k[[x]]$ and contains $R'[[Z_2]]/R'[[X]]$ as a sub-extension. By a local-to-global principle, this local assertion is sufficient for the global refined Oort conjecture and concerns extending a prescribed local lift through a cyclic Galois extension.