The general-type component conjecture for Moishezon families

Let $\mathbb D$ denote the complex disc. Let $X$ be a smooth analytic space and let $g:X\to\mathbb D$ be a proper morphism. Assume that one irreducible component of the central fiber $X_0$ is of general type.

The general-type component conjecture. Then $g$ is Moishezon, and all other fibers are of general type over a possibly smaller disc.

The source explains that smooth projective K3 and elliptic surfaces can deform to spaces that are not Moishezon, suggesting that general type may be the strongest conclusion available. A positive answer is cited when the central fiber is irreducible, projective, and canonical, but the stated general version remains open.