The Gamma-analogue of the spherical Harish-Chandra isomorphism

Let $c=(k,c')\in {\sf C}({\boldsymbol{\Gamma}}_n)$, let $I_{k,c'}$ be the invariant two-sided ideal defined from the annihilator of the representation $\varrho_{k,c'}$, and let ${\mathbf e}{\mathsf H}_{1,c}{\mathbf e}$ be the corresponding spherical symplectic reflection algebra.

Gamma spherical Harish-Chandra conjecture. There is a filtration-preserving algebra isomorphism

$${\mathcal{D}}({\mathsf{R}}Q_n)^{{\operatorname{PG}}_{\Gamma,n}}/I_{k,c'}\simeq {\mathbf e}{\mathsf H}_{1,c}{\mathbf e}.$$

The source explicitly labels this as a conjectural Gamma-analogue of the spherical Harish-Chandra isomorphism and refers to a related result as planned for a subsequent publication.