Collective infinite transitivity conjecture for Calogero–Moser spaces

Let

$$\mathcal{C}=\bigsqcup_{n\geq 0}\mathcal{C}_n$$

be the disjoint union of the $G$-spaces $\mathcal{C}_n$. For positive integers $k_1,\ldots,k_m$ and pairwise distinct indices $n_1,\ldots,n_m$, the corresponding configuration stratum is

$$\mathcal{C}_{n_1}^{[k_1]}\times\cdots\times\mathcal{C}_{n_m}^{[k_m]}.$$

An action is collectively infinitely transitive if it is transitive on every such stratum for every total number of points.

Collective infinite transitivity conjecture. The action of $G$ on $\mathcal{C}$ is collectively infinitely transitive.

This conjecture simultaneously contains the paper's transitivity and infinite-transitivity assertions for the individual Calogero–Moser spaces. It remains open.