The fiberwise birational model conjecture for Moishezon morphisms

Let $\mathbb D$ denote the complex disc. A proper morphism is Moishezon if it is bimeromorphic to a projective morphism. A morphism $g:X\to\mathbb D$ is fiberwise birational to $g^{\rm p}:X^{\rm p}\to\mathbb D$ when the corresponding fibers are birational for every point of $\mathbb D$. Assume that $g:X\to\mathbb D$ is flat, proper, and Moishezon, and that $X_0$ has canonical, respectively log terminal, singularities.

The fiberwise birational model conjecture. Then $g$ is fiberwise birational to a flat, projective morphism $g^{\rm p}:X^{\rm p}\to\mathbb D$ such that, respectively, $X_0^{\rm p}$ has canonical, respectively log terminal, singularities; $X_s^{\rm p}$ has terminal singularities for $s\neq 0$; and $K_{X^{\rm p}}$ is $\mathbb Q$-Cartier.

This conjecture concerns improving the projective model of a Moishezon morphism over a one-dimensional base so that every fiber, including the special fiber, is controlled birationally. The source introduces it as an open question in the context of projective models of Moishezon families.