Finite generation conjecture for purely log terminal pairs of log general type

Let $(X,\Delta)$ be a projective purely log terminal pair such that $\lfloor\Delta\rfloor$ is irreducible, with $\Delta$ a $\mathbb{Q}$-divisor, and suppose that $K_X+\Delta$ is big. Define

$$R(X,\Delta):=\bigoplus_{m\geq 0}H^0\left(X,\mathcal{O}_X\left(\left\lfloor m(K_X+\Delta)\right\rfloor\right)\right).$$

Finite generation conjecture. The log canonical ring $R(X,\Delta)$ is finitely generated.

This is the log-general-type purely log terminal case used in the paper to reduce finite generation for log canonical pairs to a more restricted conjecture.