Let $k$ be a finite field of characteristic $p > 0$, and let $X$ be a smooth proper variety over $k$. It is generally unknown whether $X$ admits a smooth proper lifting over $W(k$, where $W(k)$ denotes the ring of Witt vectors over $k$.
Instead of considering $W(k)$, let's take a finite extension $K/W(k)[1/p]$ and its ring of integers $\mathcal{O}_K$. My question is the following:
For a given variety $X$, does there necessarily exist a finite extension $K/W(k)[1/p]$ and a smooth proper variety $\tilde{X}$ over $\mathcal{O}_K$ such that $$ X \otimes_k \kappa(\mathcal{O}_K) \simeq \tilde{X} \otimes_{\mathcal{O}_K} \kappa(\mathcal{O}_K), $$ where $\kappa(\mathcal{O}_K)$ denotes the residue field of $\mathcal{O}_K$?
Additionally, if such a finite extension $K/W(k)[1/p]$ exists, would this extension necessarily be unramified?
