Arithmetic form of the ring-specific Oort conjecture

Set $K=\widehat{\mathbb{Q}_p^{un}}$ and $L=K(\zeta_{p^\infty})$. Let $C$ be a finite cyclic group and define

$$R_C:=R_K[\zeta_{|C|}]\subset R_L.$$

Let $M|L$ be a finite cyclic extension of $L$ with group $C$, and set

$$D:=\operatorname{Spec}(R_C[[Z]]).$$

Arithmetic ring-specific Oort conjecture. There exist $l>0$ and a normal, $C$-Galois, regular branched cover $Y\rightarrow D$ such that $Y_k$ is reduced, $L_m=M$ for $m\gg0$ and $m\equiv1\pmod l$, and $d_\eta=d_m$ for $m\gg0$ and $m\equiv1\pmod l$. This reformulates the ring-specific lifting conjecture arithmetically via the field-of-norms construction; the supplied text does not state a resolution.