Let $\mathfrak{X}$ be a locally noetherian adic formal scheme over $\text{Spf}(A)$, with $A$ an $I$-adically complete and separated noetherian ring.

Suppose the mod $I$-fiber of $\mathfrak{X}$ is an algebraic $\text{Spec}(A/I)$-scheme equipped with an ample line bundle.

Grothendieck's formal existence theorem implies $\mathfrak{X}$ is algebraizable by some $X$ over $\text{Spec}(A)$, projective, and such that $I$-adic completion gives an equivalence of categories:

$${Coh}(X)\cong {Coh}(\mathfrak{X}).$$

This equivalence implies that every closed formal subscheme $\mathfrak{Z}\subset\mathfrak{X}$ uniquely algebraizes to a closed subscheme $Z\subset X$.

QUESTION.Can we algebraize open formal subschemes of $\mathfrak{X}$ too?

For example by declaring, for any open formal subscheme $\mathfrak{U}\subset\mathfrak{X}$, that its algebraization $U$ is the open complement of the unique closed subscheme $Z$ that algebraizes $\mathfrak{Z} := \mathfrak{X}-\mathfrak{U}$, where this last one is endowed with the reduced induced formal scheme structure.