Given an algebraic fiber space $X \to B$ where $X$ and $B$ are smooth projective varieties over $\mathbb{C}$, it is known that the Kodaira dimensions satisfy the following subadditivity property: $$\kappa(X) \ge \kappa(\text{general fiber}) + \kappa(B).$$ Does the same property holds for compact Kähler manifolds? If it is still unknown, are there any partial results in this direction?
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