The finite generation conjecture for log canonical rings

Let $X$ be an $n$-dimensional smooth projective variety and let $B$ be a boundary $\mathbb Q$-divisor whose support is a simple normal crossing divisor on $X$. Assume that $K_X+B$ is big. Finite generation conjecture. The log canonical ring

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

is a finitely generated $\mathbb C$-algebra. This finite generation statement is stronger than flip existence and implies the relative finite generation needed in the log minimal model program. The source notes that it is known when a suitable klt $\mathbb Q$-divisor exists, while the general conjecture is not established there.