Parabolic restriction conjecture for invariant differential-operator quotients

Let $V$ be an $n$-dimensional vector space over $\Bbbk$, let $G=\operatorname{GL}(V)$ with Lie algebra $\mathfrak g=\mathfrak{gl}(V)$, and let $P\subset G$ be the parabolic subgroup preserving a fixed line in $V$, with Lie algebra $\mathfrak p$. For $c\in\Bbbk$, let $J_c\subset\mathcal U\mathfrak p$ be the two-sided ideal generated by $x-c\operatorname{tr}(x)$ for $x\in\mathfrak p$, and let $\operatorname{Ind}_c=\operatorname{Ind}(\mathcal U\mathfrak g\uparrow J_c)\subset\mathcal U\mathfrak g$ be the induced ideal. Let $\mathscr D(\mathfrak g)$ be the algebra of differential operators on $\mathfrak g$, with $G$ acting by the adjoint action. Parabolic restriction conjecture. For every $c\in\Bbbk$, the canonical map

$$\left(\mathscr D(\mathfrak g)/\mathscr D(\mathfrak g)\cdot\operatorname{Ind}_c\right)^G\longrightarrow\left(\mathscr D(\mathfrak g)/\mathscr D(\mathfrak g)\cdot J_c\right)^P$$

induced by the natural quotient map is an algebra isomorphism.

This conjecture asserts that taking $G$-invariants after quotienting by the induced primitive ideal agrees with taking $P$-invariants after quotienting by the original parabolic ideal. The supplied text proposes the statement but gives no evidence of a resolution.