Deformation invariance of plurigenera for canonical varieties

Let $g:X\to S$ be a flat, proper morphism of complex analytic spaces. Assume that $X_0$ is projective and has canonical singularities. For each positive integer $r$, the plurigenus is $h^0(X_s,\omega_{X_s}^{[r]})$.

Deformation invariance of plurigenera. The plurigenera $h^0(X_s,\omega_{X_s}^{[r]})$ are independent of $s\in S$ for every $r$, in some neighborhood of $0\in S$.

This would extend deformation invariance of plurigenera beyond varieties of general type. The source describes the assertion as likely true but notes difficulties in higher relative dimensions, including termination of the minimal model program and abundance; it is therefore open.