The rational-singularity limit conjecture for Moishezon fibers

Let $\mathbb D$ denote the complex disc. Let $g:X\to\mathbb D$ be a flat, proper morphism, and write $X_s$ for its fiber over $s\in\mathbb D$. Assume that $X_0$ is irreducible with rational singularities and that $X_s$ is Moishezon for every $s\neq 0$.

The rational-singularity limit conjecture. Then $X_0$ is Moishezon.

The source notes claimed positive answers in the smooth case, while the analogous statement for surfaces with cusp singularities is false. It therefore treats rational singularities as a plausible class in which the conjecture may hold, but leaves the stated version open.