Infinite transitivity conjecture for Calogero–Moser spaces

Let $G$ act on the Calogero–Moser space $4\mathcal{C}_n4$ for each integer $n\geq 1$. An action on a space $X$ is infinitely transitive if it is $k$-transitive for every positive integer $k$, where $k$-transitivity means transitivity on the configuration space

$$X^{[k]}:=\{(x_1,\ldots,x_k)\in X^k:\ x_i\neq x_j\}\,.$$

Infinite transitivity conjecture. For each $n\geq 1$, the action of $G$ on $\mathcal{C}_n$ is infinitely transitive.

The conjecture strengthens the known double-transitivity result. General results on special automorphism groups imply that transitivity of the special automorphism group is equivalent to infinite transitivity for these smooth spaces; the conjecture is known for $n=1$, while the assertion for $n\geq 2$ remains open.