Let $R$ be a complete DVR with quotient field $K$ and let $f:\mathfrak{X} \to \mathrm{Spf}(R)$ be a smooth proper formal scheme.

If the (rigid analytic) generic fibre of $f$ is (the analytification of) a projective variety $X$ over $\mathrm{Spec}(K)$, then is $f$ the formal completion of a smooth projective scheme over $\mathrm{Spec}(R)$?

One could also drop the projectivity hypothesis and just assume that the generic fibre "is" a (proper) variety over $K$.

It seems reasonable to expect this to be true at least when one knows that $X$ has a unique smooth proper model over $\mathrm{Spec}(R)$ (which is a scheme).