Log Iitaka conjecture for log canonical fibrations

Let $f\colon X\to Y$ be a surjective morphism of smooth projective varieties with connected fibers. Let $\Delta_{X}$ and $\Delta_{Y}$ be reduced simple normal crossing divisors on $X$ and $Y$, respectively, such that

$$\operatorname{Supp}f^{*}\Delta_{Y}\subset \operatorname{Supp}\Delta_{X}.$$

Let $F$ be a sufficiently general fiber of $f$, and define $\Delta_{F}$ by

$$K_{F}+\Delta_{F}=(K_{X}+\Delta_{X})|_{F}.$$

Log Iitaka conjecture. One has

$$\kappa(X,K_{X}+\Delta_{X})\geq \kappa(F,K_{F}+\Delta_{F})+\kappa(Y,K_{Y}+\Delta_{Y}).$$

This is the logarithmic analogue of the Iitaka conjecture, motivated by minimal model theory and Iitaka-type questions for morphisms between open varieties. The paper's abstract states that it proves this conjecture for log canonical fibrations when the log canonical divisor of a sufficiently general fiber is abundant; the unrestricted statement is therefore not established here.