I am looking for a proof of the following statement:

Let $π: X \to Z$ be a surjective morphism between smooth projective varieties such that $-K_X$ is nef and $Z$ is non-uniruled then Kodaira dimension of base $\kappa(Z)= 0$. What about when we replace projective varieties with "Kähler manifolds"

I am looking for a proof of the following statement:

Let $f: X \to B$ be a surjective morphism between smooth projective varieties such that $-K_X$ is nef and $B$ is non-uniruled then Kodaira dimension of base $\kappa(B)= 0$. What about when we replace projective varieties with "Kähler manifolds"

I am looking for a proof of the following statement:

Let $π: X \to Z$ be a surjective morphism between smooth projective varieties such that $-K_X$ is nef and $Z$ is non-uniruled then Kodaira dimension of base $\kappa(Z)= 0$.

I am looking for a proof of the following statement:

Let $π: X \to Z$ be a surjective morphism between smooth projective varieties such that $-K_X$ is nef and $Z$ is non-uniruled then Kodaira dimension of base $\kappa(Z)= 0$. What about when we replace projective varieties with "Kähler manifolds"