Reduction conjecture for equivariant and quotient quantum D-modules

Let $X$ be a smooth projective variety equipped with an algebraic $T_\mathbb{C}$-action, and let $Y$ be a smooth GIT quotient of $X$ without orbifold singularities. Let $J_X=J_X(\tau,z)$ be the big equivariant $J$-function and let $\kappa\colon H_T^*(X)\to H^*(Y)$ be the Kirwan map. For $\beta\in N_1^T(X)$, write $\widehat{S}^{\beta}$ for the corresponding element of the group ring $\mathbb{C}[N_1^T(X)]$, and define

$$I_Y:=\sum_{[\beta]\in N_1^T(X)/N_1(T)}\kappa\bigl(\widehat{\mathcal{S}}^{-\beta}J_X\bigr)\widehat{S}^{\beta} \in\sum_{\beta\in N_1^T(X)}H^*(Y)[z,z^{-1}][[\tau]]\widehat{S}^{\beta}.$$

Here each $\beta$ is a representative of its class, and each summand is independent of that choice. Let $C_Y\subset N_T^1(X)_\mathbb{R}$ be the GIT chamber of $Y$, and let $C_{Y,\mathbb{N}}^\vee$ consist of those $\beta\in N_1^T(X)$ whose image lies in the dual cone $C_Y^\vee$. Reduction conjecture. The series $I_Y$ is supported on $C_{Y,\mathbb{N}}^\vee$, and $zI_Y$ is a point on the non-equivariant Givental cone $\mathcal{L}_Y$ of $Y$, defined over the extension $\mathbb{C}[[C_{Y,\mathbb{N}}^\vee]]$ of the Novikov ring of $Y$. This gives a solution-level formulation of the relationship between the equivariant quantum $D$-module of $X$ and the quantum $D$-module of its GIT quotient; it is presented as a more rigorous version of the preceding informal reduction conjecture. The source gives no resolution status.