Kawamata's relative Fujita conjecture

Let $f:Y\to X$ be a proper morphism between smooth projective algebraic varieties. Assume that the degenerate locus of $f$ is contained in a normal crossing divisor $D\subset X$. Let $L$ be an ample line bundle on $X$. Kawamata's conjecture. For every $0\leq q\leq \dim_\mathbb{C}X$, the sheaf

$$R^qf_*\omega_Y\otimes L^{\dim X+1}$$

is generated by global sections.

This is a relative version of Fujita's conjecture; the case $\dim_\mathbb{C}X\leq 4$ was settled by Kawamata, while the statement in general remains open.