Boundedness conjecture for K-semistable Fano cone singularities

Let $n$ be a positive integer. A set of klt singularities is bounded if all its members occur as fibers of one $\mathbb{Q}$-Gorenstein family; for Fano cone singularities, the family also carries a compatible fiberwise torus action containing the Reeb vectors. Boundedness conjecture. For every real number $\varepsilon>0$, the set of $n$-dimensional K-semistable Fano cone singularities $x\in(X;\xi)$ satisfying

$$\widehat{\rm vol}(x,X)\geq\varepsilon$$

is bounded. This is the local analogue of boundedness in K-moduli theory and would support moduli spaces for K-polystable Fano cone singularities; the source says boundedness remains mysterious, although criteria and some low-dimensional cases are known.