Finite generation for valuations computing the stability threshold

Let $(X,\Delta)$ be a log Fano variety with $\delta(X,\Delta)\le 1$, and let $r$ be a positive integer such that $-r(K_X+\Delta)$ is Cartier. Define the section ring

$$R=\bigoplus_{m\in\mathbb{N}}H^0\bigl(-mr(K_X+\Delta)\bigr).$$

If a valuation $v$ computes $\delta(X,\Delta)$, then

Finite-generation conjecture. The associated graded ring $\operatorname{gr}_v R$ is finitely generated.

This is the global variant of the finite-generation conjecture for normalized-volume minimizers. It concerns valuations computing the stability threshold of a log Fano pair and remains open in general.