Saidi's Oort conjecture revisited on lifting cyclic extensions in towers

Let $k$ be an algebraically closed field of characteristic $p$. Consider a local $G$-extension

$$k[[z]]/k[[t]]$$

with $G$ cyclic. Let

$$R[[S]]/R[[T]]$$

be a characteristic-zero lift of a subextension

$$k[[s]]/k[[t]].$$

A lift of $k[[z]]/k[[t]]$ is a characteristic-zero extension of complete local power series rings reducing to the given extension.

Oort conjecture revisited. Local cyclic extensions are liftable in towers: given the local $G$-extension $k[[z]]/k[[t]]$ and a characteristic-zero lift $R[[S]]/R[[T]]$ of a subextension $k[[s]]/k[[t]]$, there is a characteristic-zero lift of $k[[z]]/k[[t]]$ containing $R[[S]]/R[[T]]$ as a subextension.

This conjecture asks for compatibility of characteristic-zero lifts with prescribed lifted subextensions, strengthening the ordinary lifting assertion for cyclic local extensions. The source attributes the formulation to Saidi and does not state a resolution.