Finite generation of multigraded adjoint rings

Let $X$ be a projective variety, let $\ell$ be a positive integer, and let

$$D_i=k_i(K_X+\Delta_i)\in\operatorname{Div}(X),\qquad i=1,\dots,\ell,$$

where each $k_i$ is as in the source and $(X,\Delta_i)$ is a log canonical pair. Define the adjoint ring by

$$R(X;D_1,\dots,D_\ell)=\bigoplus_{(m_1,\dots,m_\ell)\in\mathbb{N}^{\ell}}H^0\bigl(X,\mathcal{O}_X(m_1D_1+\cdots+m_\ell D_\ell)\bigr).$$

Adjoint-ring finite-generation conjecture. The adjoint ring $R(X;D_1,\dots,D_\ell)$ is finitely generated.

The source presents this conjecture as potentially encoding the main theorems of Mori theory, including existence and termination of flips and abundance, when applied to Mori dream regions. Its status is not resolved in the supplied text.