Oort's lifting conjecture for local cyclic extensions

Let $S$ be the fixed field of $\ker(\pi)$, identified with $k((\overline{u}))$ for a parameter $\overline{u}$. The action of $G_F$ defines a continuous $k$-linear action $\tau$ of $\mu_{p^n}$ on $k \llbracket \overline{u} \rrbracket$. A solution of the lifting problem for $\pi$ is a lifting of $\tau$ to a continuous $\mathfrak{o}_K$-linear action $\tilde{\tau}$ of $\mu_{p^n}$ on $\mathfrak{o}_K \llbracket u \rrbracket$. Oort's conjecture. A solution of the lifting problem exists for every $\pi$. This is the local lifting problem for cyclic extensions of degree a power of $p$, asking whether the characteristic-$p$ action can always be lifted to characteristic zero. The source provides no resolution evidence, so the conjecture is recorded as open.