Kawamata's conjecture on global generation of direct images of canonical bundles

Let $f:Y\rightarrow X$ be a surjective morphism from a smooth projective variety to a smooth projective variety of dimension $n$, such that $f$ is smooth over

$$X_0=X\setminus B,$$

where $B$ is a normal crossing divisor on $X$. Let $L$ be an ample divisor on $X$.

Kawamata's conjecture. The locally free sheaf

$$R^jf_\ast\omega_Y\otimes L^{\otimes m}$$

is generated by global sections if $m\geq n+1$, or if $m=n$ and the intersection number

$$(L^n)\geq 2,$$

for all $j\geq 0$.

This is proposed as a relative Fujita-type statement for higher direct images of canonical bundles. The supplied text does not give a resolution of the conjecture.