Deformation invariance of semiampleness and generalized plurigenera

Let $\pi:\mathcal{X}\rightarrow \Delta$ be a smooth family of compact Kähler manifolds, or even of compact complex manifolds in Fujiki class $\mathcal{C}$, meaning manifolds bimeromorphic to compact Kähler manifolds. Write $K_{X_t}$ for the canonical line bundle of the fiber over $t$, and let $K_{\mathcal{X}}$ denote the canonical line bundle of the total space. A line bundle is semiample if some positive tensor power is globally generated, and $K_{\mathcal{X}}$ is $\pi$-semiample over an open subset if such a power is generated relative to that subset. If $K_{X_0}$ is nef, then

Deformation invariance conjecture. For every $t\in\Delta$, $K_{X_t}$ is semiample and $K_{\mathcal{X}}$ is $\pi$-semiample over $\pi^{-1}(U_t)$, where $U_t$ is a Zariski neighborhood of $t$. Furthermore, for every $i\geq 0$ and $m\geq 1$, the generalized $m$-genus $P^i_m(X_t)$ is independent of $t\in\Delta$.

This proposes extending the corresponding stability of semiampleness and deformation invariance of generalized plurigenera known for smooth Kähler families of threefolds to arbitrary dimensions and to families in Fujiki class $\mathcal{C}$. The proposed extension remains open in the stated generality.