Kollár's canonical model conjecture after a base change

Let $g:\mathcal X\to\Delta$ be a flat, proper, Moishezon morphism, where $\Delta$ is the unit disc in $\mathbb C$. Assume that $X_0$ has only canonical singularities. A morphism $\nu:\widetilde\Delta\to\Delta$ is surjective if it is onto, and let $\widetilde g$ denote the base change of $g$ induced by $\nu$. Kollár's base-change canonical model conjecture. After shrinking $\Delta$, there exists a surjective morphism $\nu:\widetilde\Delta\to\Delta$ with $\tau\in\nu^{-1}(0)$ such that $\widetilde g$ is fiberwise bimeromorphic, possibly after shrinking $\widetilde\Delta$, to a flat, projective morphism

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

whose fiber over $\tau$ has only canonical singularities, whose fibers over $s\ne\tau$ have terminal singularities, and for which $K_{\mathcal X^{\mathrm P}}$ is $\mathbb Q$-Cartier. This is proposed as a possible approach to Kollár's canonical-case conjecture; the supplied text gives no resolution.