Viehweg's conjecture on the strengthened subadditivity of Kodaira dimension

Let $f\colon X \to Y$ be a fibration between smooth projective varieties with general fiber $F$ and $\kappa(Y)\ge0$. Define $\operatorname{Var}(f)$ to be the variation of the fibration. Viehweg's conjecture.

$$\kappa(X)\ge\kappa(F)+\max\{\kappa(Y),\,\operatorname{Var}(f)\}.$$

This strengthens the Iitaka conjecture by incorporating the variation of the fibration. It is known in a few cases, including when $Y$ is of general type, $F$ has a good minimal model, or $F$ is of general type, and remains open in general.