Finite generation of the canonical algebra near rational surface sections

Let $X$ be a three-dimensional normal singularity, let $H\subset X$ be a Cartier divisor containing a point $x$, and assume that $(x\in H)$ is a normal rational surface singularity. Consider the canonical algebra

$$\bigoplus_{m\geq 0}\mathcal{O}_X(mK_X).$$

Finite-generation conjecture. The sheaf $\bigoplus_{m\geq 0}\mathcal{O}_X(mK_X)$ is a finitely generated sheaf of $\mathcal{O}_X$-algebras.

The source says that this conjecture is known when $x\in H$ is a quotient singularity or a quadruple point, but gives no resolution in the stated generality. It is presented as apparently distinct from the minimal model program outside the quotient-singularity case.