Ueno's Conjecture K on Albanese maps

Let $X$ be a nonsingular projective variety with $\kappa(X)=0$. Ueno's Conjecture K asserts that the Albanese map is birational to an étale fiber bundle over $\operatorname{Alb}(X)$ which is trivialized by an étale base change. This is a structural conjecture about varieties of Kodaira dimension zero; the supplied passage gives no resolution, so it remains open here.