b-semi-ampleness conjecture for analytic lc-trivial fibrations

Let $X$ be a normal complex analytic variety in Fujiki's class $\mathcal C$, let $\Delta$ be a $\mathbb Q$-divisor such that $K_X+\Delta$ is $\mathbb Q$-Cartier, and let $f:X\to Y$ be a surjective morphism onto a normal projective variety such that $f:(X,\Delta)\to Y$ is an lc-trivial fibration. Let $\mathbf M$ be its moduli $\mathbb Q$-b-divisor. Analytic b-semi-ampleness conjecture. The moduli b-divisor $\mathbf M$ is b-semi-ample. The algebraic analogue is presented as a major open problem, and the source says this analytic version may be harder because good moduli theory for compact Kähler manifolds is unavailable.