The good-reduction lifting conjecture for cyclic covers of the projective line

Let $K$ be the ground field, let $X=\mathbb{P}^1_K$, and let $\bar{x}_0$ be the distinguished branch point with residue field $\kappa_0$. For a character $\chi\in H^1_{p^n}(\mathbb{K})$, call it admissible if it defines the relevant $\mathbb{Z}/p^n$-cover, and call it a lift of $\bar{\chi}\in H^1_{p^n}(\kappa_0)$ when its reduction is $\bar{\chi}$. Say that such a character has good reduction when it has étale reduction and the cover corresponding to its reduction is smooth.

Suppose that $X=\mathbb{P}^1_K$. Let $\bar{\chi}\in H^1_{p^n}(\kappa_0)$ be a character of order $p^n$, unramified outside $\bar{x}_0$. Then, after replacing $K$ by a finite extension if necessary, there exists an admissible character $\chi\in H^1_{p^n}(\mathbb{K})$ with good reduction lifting $\bar{\chi}$.

This is a reformulation of the local Oort conjecture for $\mathbb{Z}/p^n$-covers of the projective line. Its status depends on the local lifting result being used; the supplied text gives no resolution evidence for this formulation.