Finiteness conjecture for Cox rings of log Fano varieties

Let $X$ be a log Fano variety. Its Cox ring $\mathrm{Cox}(X)$ is the ring graded by $\mathrm{Pic}(X)$ associated with a basis of the Picard group as defined above.

Finiteness of Cox ring. The ring

$$\operatorname{Cox}(X)$$

is finitely generated.

Finite generation of the Cox ring implies finite generation of the cone of effective divisors and, via the torus action on the Cox ring, finite generation of the nef cone. The source presents this as a special case of a conjecture in the cited literature; no resolution status is supplied here.