Global Nakano positivity conjecture for direct images with multiplier ideals

Let $f\colon X\to Y$ be a projective surjective morphism between complex manifolds. Let $(L,h)$ be a holomorphic line bundle over $X$ equipped with a singular Hermitian metric of semi-positive curvature, and let $K_{X/Y}$ denote the relative canonical bundle and $\mathscr{I}(h)$ the multiplier ideal sheaf of $h$. Global Nakano positivity conjecture. The pushforward sheaf

$$f_\star\bigl(K_{X/Y}\otimes L\otimes\mathscr{I}(h)\bigr)$$

admits a canonical singular Hermitian metric that is globally Nakano semi-positive in the sense of the paper's definition. This conjecture proposes a global singular analogue of Berndtsson's Nakano semi-positivity theorem for direct images; the corresponding local result is established in the paper, while the asserted global construction and positivity remain open.