Finite-generation conjecture for log canonical rings in Fujiki's class

Let $X$ be a normal complex analytic variety in Fujiki's class $\mathcal C$ and let $\Delta$ be an effective $\mathbb Q$-divisor on $X$ such that $(X,\Delta)$ is log canonical. Its log canonical ring is

$$R(X,\Delta)=\bigoplus_{m\geq 0}H^0(X,\mathcal O_X(\lfloor m(K_X+\Delta)\rfloor)).$$

Fujiki-class finite-generation conjecture. The ring $R(X,\Delta)$ is a finitely generated $\mathbb C$-algebra. The source states that this conjecture is equivalent, after taking a resolution, to the preceding compact Kähler formulation and remains open even in the projective case.