Semipositivity conjecture for relative pluricanonical direct images

Let $f:X\to Y$ be a surjective morphism between smooth projective varieties with connected fibers. Then there exists a generically finite morphism $\tau:Y'\to Y$ from a smooth projective variety $Y'$ with the following property. Let $X'$ be any resolution of the main component of $X\times_Y Y'$ sitting in the commutative diagram

\xymatrix{X' \ar[r]\ar[d]_{f'}& X\ar[d]^f\\ Y'\ar[r]_{\tau} &Y. }

Semipositivity conjecture for relative pluricanonical direct images. The sheaf $f'_*\omega^{\otimes m}_{X'/Y'}$ is nef and locally free for every positive integer $m$.

This extends the known semipositivity result for the relative canonical direct image ($m=1$) to all positive pluricanonical powers; the source does not state whether the conjecture has been resolved.