The Harish-Chandra-type conjecture for rational Cherednik algebras

Let $V$ be the defining representation, let $B$ be its Borel subgroup, let $\mathfrak b$ be the Lie algebra of $B$, and put $H=B\cap SL(V)$. Let $\mathbb P$ denote the projective space appearing in the algebra of twisted differential operators, let $\mathcal D(\mathfrak b\times\mathbb P,c)$ be the algebra of $c$-twisted differential operators on $\mathfrak b\times\mathbb P$, and define

$$\mathfrak b_c=\operatorname{Im}\bigl(\mathfrak b\cap\mathfrak{sl}(V)\longrightarrow\mathcal D(\mathfrak b\times\mathbb P,c)\bigr).$$

Let $H_{1,\mathbf c}$ be the rational Cherednik algebra at parameter $t=1$, let $e$ be its spherical idempotent, and let $\operatorname{ad}\mathfrak b_c$ denote the adjoint action. The associated graded algebra is taken with respect to the natural filtration.

Harish-Chandra-type conjecture. There exists a subalgebra $H_{1,\mathbf c}'\subseteq H_{1,\mathbf c}$ such that

$$\left(\mathcal D(\mathfrak b\times\mathbb P,c)/\mathcal D(\mathfrak b\times\mathbb P,c)\cdot\mathfrak b_c\right)^{\operatorname{ad}\mathfrak b_c}\stackrel{\simeq}{\longrightarrow}eH_{1,\mathbf c}'e$$

and

$$\operatorname{gr}\left(\mathcal D(\mathfrak b\times\mathbb P,c)/\mathcal D(\mathfrak b\times\mathbb P,c)\cdot\mathfrak b_c\right)^{\operatorname{ad}\mathfrak b_c}\stackrel{\simeq}{\longrightarrow}\mathbb C[V]^H.$$

This is proposed as an analogue of classical Harish-Chandra homomorphisms, relating adjoint invariants of a quotient of twisted differential operators to a spherical subalgebra of a rational Cherednik algebra and identifying its associated graded algebra with the invariant ring $\mathbb C[V]^H$. No resolution is supplied in the given text.