Global versal-family conjecture for proper rigid analytic spaces

Let $X$ be a proper rigid analytic space over $K$. A formal versal deformation of $X$ is the formal deformation supplied by the local deformation theory of $X$. Global versal-family conjecture. There exists a flat morphism of rigid analytic spaces

$$F: \mathfrak{X} \longrightarrow \mathfrak{S}$$

and a $K$-rational point $p$ of $\mathfrak{S}$ such that the completion of $F$ at $p$ is isomorphic to the formal versal deformation of $X$. This is intended as a non-Archimedean analogue of the existence of versal families for complex analytic spaces. The conjecture is proved for rigid analytic tori using $p$-adic uniformization, but the source states that it is unproved for general rigid analytic spaces; the proposed sufficient condition involves a locally noetherian adic formal model over the ring of integers of a discretely valued field.