Local freeness conjecture for pluricanonical direct images of weakly semistable morphisms

Let $f:X\to Y$ be a weakly semistable morphism with connected fibers. Here weak semistability includes the toroidal, equidimensional, reduced-fiber, and smooth-base conditions stated in the source.

Local freeness conjecture for weakly semistable morphisms. The sheaf

$$f_*\omega^{\otimes m}_{X/Y}$$

is locally free for every $m\geq 1$.

This is proposed as a conjecture for weakly semistable morphisms and concerns the local behavior of relative pluricanonical direct images; the source does not state whether it has been resolved.