The generalized Kähler canonical bundle formula conjecture

Let $f:X\to Z$ and $Z\to S$ be proper morphisms of normal relatively compact Kähler varieties such that $f_*\mathcal O_X=\mathcal O_Z$, and let $(X/S,B+\boldsymbol\beta)$ be a generalized pair. Suppose that

$$K_X+B+\boldsymbol\beta_X=f^*\gamma$$

for a $\overline{\partial},\partial$-closed current $\gamma$ on $Z$. Let $\boldsymbol\gamma=\overline\gamma$, define the boundary b-divisor $\mathbf B^Z$ and the moduli part $\boldsymbol\beta^Z$ by

$$\boldsymbol\beta^Z=\boldsymbol\gamma-(\mathbf K+\mathbf B^Z),\qquad K_{Z'}+\mathbf B^Z_{Z'}+\boldsymbol\beta^Z_{Z'}=\boldsymbol\gamma_{Z'}.$$

Here $B_Z=\mathbf B^Z_Z$ on the given model.

Generalized Kähler canonical bundle formula conjecture. If $f:(X,B+\boldsymbol\beta)\to Z$ is a generalized klt (respectively, lc) pair as above, then $(Z,B_Z+\boldsymbol\beta^Z)$ is a generalized klt (respectively, lc) pair.

This is the expected singularity statement in the canonical bundle formula for generalized Kähler pairs, analogous to the algebraic case. The source also expects the discriminant and moduli b-divisors to descend to a model and the moduli part to be nef, but those additional expectations are not part of the displayed conjecture. The resolution status is not specified in the source.