Kollár's fiberwise bimeromorphic model conjecture for canonical families

Let $g:\mathcal X\to\Delta$ be a flat, proper, Moishezon morphism over the unit disc $\Delta a$ in $\mathbb C$. Assume that $X_0$ has only canonical singularities. A morphism is fiberwise bimeromorphic to another morphism when the corresponding fibers are bimeromorphic. Kollár's fiberwise bimeromorphic model conjecture. The morphism $g$ is fiberwise bimeromorphic to a flat, projective morphism

$$g^{\mathrm p}:\mathcal X^{\mathrm p}\to\Delta$$

possibly over a smaller disc, such that $X_0^{\mathrm p}:=(g^{\mathrm p})^{-1}(0)$ has only canonical singularities, $X_s^{\mathrm p}:=(g^{\mathrm p})^{-1}(s)$ has terminal singularities for $s\ne0$, and $K_{\mathcal X^{\mathrm P}}$ is $\mathbb Q$-Cartier. Kollár proposed this conjecture as an existence statement for a fiberwise projective model; the supplied text says that the log-terminal case has a positive answer under an additional non-uniruledness assumption, while the canonical case remains under discussion.