J-function conjecture for nonabelian quotients

Let $G$ act on a smooth projective variety $X$, let $T$ be a maximal torus of $G$, and write $X//G$ and $X//T$ for the corresponding quotients. For $d\in H_2(X//G)$, define

$$I_d(\hbar):=\sum_{\widetilde d\mapsto d}\left(\prod_{\alpha}\frac{\prod_{k=-\infty}^{\widetilde d\cdot c_1(L_{\alpha})}(c_1(L_{\alpha})+k\hbar)}{\prod_{k=-\infty}^{0}(c_1(L_{\alpha})+k\hbar)}\right)J^{X//T}_{\widetilde d}(\hbar),$$

where the sum is over curve classes $\widetilde d\in H_2(X//T)$ lifting $d$, and set

$$I(t_0,\mathbf t,\hbar)=e^{t_0+\mathbf t/\hbar}\sum_d e^{\int_d\mathbf t}I_d(\hbar).$$

J-function conjecture. $J^{X//G}(t_0,\mathbf t,\hbar)$ is obtained from $I$ by an explicit change of variables (the “mirror transformation”). If $X//G$ is Fano of index at least $2$, then

$$J^{X//G}(t_0,\mathbf t,\hbar)=I(t_0,\mathbf t,\hbar).$$

The conjecture gives a generating-function form of the nonabelian/abelian correspondence. The paper states that it proves the $J$-function conjecture for flag manifolds, so the displayed assertion is not uniformly open in that proved special case; the supplied parser status gives no basis for a broader resolution.