Fujita-number subadditivity conjecture for smooth fibrations

Let $X$ and $Y$ be smooth projective varieties, let $\pi\colon X\to Y$ be a smooth fibration, and let $f_X$, $f_Y$, and $f_\pi$ denote the corresponding Fujita numbers. Fujita-number subadditivity conjecture. One should have

$$f_X\leq f_\pi\cdot f_Y.$$

The conjecture is motivated by analogies with the subadditivity of Kodaira dimension and by examples including surfaces, products of varieties with no non-trivial correspondences, and the results proved in the paper. The source does not provide evidence of a resolution, so the conjecture remains open.