Deformations over $A_{\inf}$

Question

Setup:

1. Let $K$ be a perfectoid field of characteristic $0$ with tilt $K^{\flat}$.
2. Let $A_{\inf}=W(\mathcal{O}_{K^{\flat}})$ be the infinitesimal period ring.
3. Let $\mathcal{X}$ be a flat, projective $\mathcal{O}_K$-scheme of finite type with smooth generic fiber $\mathcal{X}_K$.
4. Recall that there exists a primitive degree 1 element $\xi$ such that $\mathcal{O}_K=A_{\inf}/(\xi)$. Write $\theta:A_{\inf}\to \mathcal{O}_K$ for the quotient map.

Question: Does there exist a flat, projective $A_{\inf}$-scheme $\mathfrak{X}$ such that $\mathcal{X}$ is the pullback of $\mathfrak{X}$ via $\theta$?

This may be too general to be true. Are there further conditions which guarantee the existence of such a deformation? Does it help if the generic fiber $\mathcal{X}_K$ is nice, e.g., isomorphic to a Fano complete intersection?