Deformation invariance of plurigenera under a globally generated pluricanonical bundle

Let $X$ be a complex space and let $g:X\to S$ be a flat, proper morphism. Assume that $X_0$ is projective and has canonical singularities, and that $\omega_{X_0}^{[r]}$ is globally generated for some $r>0$.

Deformation invariance under global generation. The plurigenera are locally constant at $0\in S$.

This is a proposed partial extension of deformation invariance of plurigenera to varieties that are not necessarily of general type. The source presents it as an open assertion arising after assuming the existence of a good minimal model.