Finite generation conjecture for log canonical pairs over complex analytic spaces

Let $\pi\colon X\to Y$ be a projective morphism of complex analytic spaces and let $\Delta$ be an effective $\mathbb Q$-divisor on $X$ such that $(X,\Delta)$ is log canonical. Define the relative log canonical ring by

$$R(X/Y,K_X+\Delta):=\bigoplus_{m\in\mathbb N}\pi_*\mathcal O_X(\lfloor m(K_X+\Delta)\rfloor).$$

Finite generation conjecture. The graded $\mathcal O_Y$-algebra $R(X/Y,K_X+\Delta)$ is locally finitely generated.

This is presented as one of the main goals of the minimal model theory for projective morphisms between complex analytic spaces. The supplied context does not state whether finite generation is known or open in the generality above.