Kivinen's hyper-Kähler equivalence conjecture for Calogero–Moser spaces

Let $\mathfrak{g}$ be a reductive Lie algebra and let $\mathcal{CM}_c=\operatorname{Spec}Z(H^{\mathrm{rat}}_{c,0})$ be its Calogero–Moser space, with $c$ as in the source. Let $X_{\mathfrak{g},\mathrm{symb}}$ be the symbolic replacement variety, equipped with the indicated Hamiltonian $\mathbb{C}^*$-action. Kivinen's hyper-Kähler equivalence conjecture. For any $\mathfrak{g}$, the varieties $\mathcal{CM}_c$ and $X_{\mathfrak{g},\mathrm{symb}}$ admit hyper-Kähler structures such that there is a $U(1)$-equivariant homeomorphism

$$\mathcal{CM}_c\to X_{\mathfrak{g},\mathrm{symb}},$$

given by rotating the complex structure. In particular, there is a bijection

$$(X_{\mathfrak{g},\mathrm{symb}})^{\mathbb{C}^*}\leftrightarrow\mathcal{CM}_c^{\mathbb{C}^*},$$

where the action on the left is the Hamiltonian $\mathbb{C}^*\subseteq(\mathbb{C}^*)^2$-action. This is motivated by the known type $A$, $B$, and $C$ cases, but is conjectural for arbitrary $\mathfrak{g}$.