The center conjecture for infinitesimal Cherednik algebras of symplectic type

Let $H_\zeta(\mathfrak{sp}_{2n})$ be the infinitesimal Cherednik algebra associated with $\mathfrak{sp}_{2n}$, and let $t_i$ be the elements defined from the symmetrizations of the invariant polynomials of degrees $2i$, for $1\leq i\leq n$. Write $\mathfrak{z}(U(\mathfrak{sp}_{2n}))$ and $\mathfrak{z}(H_\zeta(\mathfrak{sp}_{2n}))$ for the centers of the enveloping algebra and the Cherednik algebra, respectively.

The center conjecture. The center of $H_\zeta(\mathfrak{sp}_{2n})$ is

$$\mathfrak{z}(H_\zeta(\mathfrak{sp}_{2n}))=\mathbb{C}[t_1+C_1,\ldots,t_n+C_n]$$

for some $C_i\in\mathfrak{z}(U(\mathfrak{sp}_{2n}))$.

This describes the center through deformations of the canonical elements $t_i$. The conjecture was recently proved using another presentation of $H_\zeta(\mathfrak{sp}_{2n})$.