The superadditivity conjecture for smooth loci of algebraic fiber spaces

Let $f\colon X\to Y$ be an algebraic fiber space between smooth projective varieties, meaning a surjective morphism with connected fibers, and let $F$ be its general fiber. Let $V\subseteq Y$ be the open subset over which $f$ is smooth. The log Kodaira dimension of $V$ is defined, after choosing a smooth projective compactification with simple normal crossing boundary $D=Y\smallsetminus V$, by

$$\kappa(V):=\kappa\bigl(Y,\omega_Y(D)\bigr).$$

The superadditivity conjecture. One should have

$$\kappa(F)+\kappa(V)\geq\kappa(X).$$

This is a proposed counterpart to Iitaka's subadditivity conjecture. It is known in several cases, including when $V$ is of log general type and for various families with maximal variation; the general case remains open.