Semiampleness conjecture for the generalized Hodge line bundle on KSBA spaces

Let $f\colon (X,\Delta+\epsilon B)\to S$ be a family of KSBA-stable log Calabi–Yau pairs with $0<\epsilon\ll 1$, where generically $K_X+\Delta\sim_{\mathbb Q}0$ and $B$ is $\mathbb Q$-Cartier and ample. Let $\lambda$ be the generalized Hodge $\mathbb Q$-line bundle defined by

$$K_{X/S}+\Delta=f^*(\lambda).$$

Semiampleness conjecture. The generalized Hodge line bundle $\lambda$ is semiample.

The preceding discussion shows that $\lambda$ is nef, because it is obtained as the leading term of the nef classes $\kappa_1(\epsilon)$ as $\epsilon\to 0$. The conjecture asks for the stronger semiampleness property on the relevant KSBA moduli stack or coarse moduli space.