Ueno's Conjecture K on varieties of Kodaira dimension zero

Let $X$ be a smooth projective variety with $\kappa(X)=0$, and let $\alpha:X\rightarrow A$ be its Albanese map, with general fibre $F$. Ueno Conjecture K. The map $\alpha$ is surjective, $\kappa(F)=0$, and there is an étale cover $A'\rightarrow A$ such that

$$X\times_A A'$$

is birational to $F\times A'$ over $A$. The first assertion is proved by Kawamata, and the second follows from work cited in the source; the birational product assertion remains open.