The moduli-part freeness conjecture

Let $f:X\to Y$ be a contraction of normal varieties and let $K_X+B$ be a log divisor such that

• $K_X+B\sim_{\mathbb{Q},f}0$;
• $K_X+B$ is Kawamata log terminal over $\eta_Y$;
• $(X,B)$ has log nonsingular support and $\mathcal O_{Y,\eta_Y}=(f_*\mathcal O_X(\lceil-B\rceil))_{\eta_Y}$.

Positivity conjecture. Then $K_Y+B_Y$ is $\mathbb{Q}$-Cartier and the moduli part $M_Y$ is $\mathbb{Q}$-free: for some $\nu\in\mathbb{N}$, $\mathcal O_Y(\nu M_Y)$ is generated by global sections. This is the anticipated positivity statement for the moduli part in the canonical bundle formula, strengthening nefness to freeness. The source attributes the expectation to Kawamata and Mori; the supplied text gives no resolution status.