The local Moishezon morphism criterion

A proper morphism of analytic spaces $g:X\to S$ is called locally Moishezon when it is locally bimeromorphic over $S$ to a projective morphism. A complex analytic space is Moishezon when it is bimeromorphic to a projective variety. For each $s\in S$, let $X_s$ denote the fiber of $g$.

The local Moishezon morphism criterion. The morphism $g$ is locally Moishezon if and only if every irreducible component of every fiber $X_s$ is Moishezon.

The paper identifies this as a main open problem, with only partial results known, including the smooth case. The claim is stated without assuming that $g$ is flat or that $X$ and $S$ are smooth.