Deformability in towers for cyclic extensions

Let $n\geq m\in\mathbb{Z}_{\geq 0}$, and consider a tower of Galois extensions

$$k[[x]]\xrightarrow{\phi_m}k[[y_m]]\xrightarrow{\phi_{n/m}}k[[y_n]],\qquad \phi_n=\phi_{n/m}\circ\phi_m.$$

Write $R_{\phi_m}$, $R_{\phi_{n/m}}$, and $R_{\phi_n}$ for the corresponding deformation rings, and let $G:=\operatorname{Gal}(k[[y_n]]/k[[x]])$.

Deformability in towers conjecture. If $G$ is cyclic, then the natural morphism

$$\operatorname{ind}:\operatorname{Spec}R_{\phi_n}\longrightarrow\operatorname{Spec}R_{\phi_m}$$

is surjective. Equivalently, for every complete discrete valuation ring $R$ with residue field $k$, after a finite extension $R'$ of $R$ one can fill in the commutative deformation diagram in the source, and in this case $\phi_n$ is called deformable in towers.

This conjecture asks whether every deformation of the lower-level cyclic cover can be lifted to the upper level after a finite extension of the base DVR. It is a proposed generalization of known descriptions of deformation rings in certain wildly ramified cases.