Let $K$ be a $p$-adic field whose residue field $k$ is algebraically closed. Let $X$ be a hyperbolic curve over $K$, what I mean by a curve is a smooth geometrically connected scheme of dimension one over $K$.

Let us further assume that there exists a flat proper morphism $\overline{\mathcal{X}}\to\mathrm{Spec}\mathcal{O}_K$ such that $\overline{\mathcal{X}}_K\times_K\overline{K}$ is the smooth completion of $X\times_K\overline{K}$, where $\overline{\mathcal{X}}_K$ is the generic fibre of $\overline{\mathcal{X}}\to\mathrm{Spec}\mathcal{O}_K$ as usual.

Now, here is the question:

**Q.** Let $\overline{\mathcal{X}}_k$ be the special fibre of $\overline{\mathcal{X}}\to\mathrm{Spec}\mathcal{O}_K$.Then, does any finite etale covering of $\overline{\mathcal{X}}_k$ lift to a finite etale covering of $\overline{\mathcal{X}}$?