Popa's superadditivity conjecture for logarithmic Kodaira dimension

Let $f:X\to Y$ be a smooth projective morphism with connected fibers between smooth quasi-projective varieties. Write $\bar\kappa$ for logarithmic Kodaira dimension, and let $F$ be the general fiber of $f$. Popa's superadditivity conjecture. One has

$$\bar\kappa(X)=\bar\kappa(Y)+\kappa(F).$$

This conjecture asserts additivity of logarithmic Kodaira dimension for smooth projective families with connected fibers; the paper proves the relevant result in the setting discussed around the statement, but the parser supplies no resolution status for the conjecture in full generality.