The finite-generation conjecture for valuations computing the delta invariant

Let $X$ be a ${\mathbb{Q}}$-Fano variety with $\delta(X)\le 1$, and let $v$ be a quasi-monomial valuation computing $\delta(X)$. Let $R$ be the relevant section ring and ${\rm gr}_vR$ its associated graded ring.

Finite-generation conjecture. The graded ring ${\rm gr}_vR$ is finitely generated.

The source explains that proving this would produce a divisorial valuation computing the delta invariant and would imply the preceding conjecture about special test configurations. The rational-rank-one case is stated to follow from BCHM, while the higher rational-rank case remains open.