Finite generation of adjoint divisorial rings

Let $X$ be a smooth projective variety, and let $B_1,\dots,B_k$ be $\mathbb{Q}$-divisors on $X$ such that $0\leq B_i<1$ for all $i$, with the support of $\sum_{i=1}^k B_i$ having simple normal crossings. Set

$$D_i=K_X+B_i$$

for every $i$, and write

$$R(X;D_1,\dots,D_k)=\bigoplus_{(m_1,\dots,m_k)\in\mathbb{N}^k}H^0\left(X,\mathcal{O}_X\left(\sum_{i=1}^k m_iD_i\right)\right).$$

Finite-generation conjecture. The ring $R(X;D_1,\dots,D_k)$ is finitely generated.

Finite generation of these adjoint divisorial rings is a central input in the minimal model program, since it leads to the existence of canonical models and supports the termination of birational surgeries. The statement is proved in the cited work of Birkar, Cascini, Hacon, and McKernan, so it is treated as solved.