The equivariant mirror conjecture for quantum differential equations of $G/P$

Let $G/P$ be a partial flag variety. Let $\tilde Z_P^{\mathfrak h^\vee}$ be the pulled-back covering mirror family, let $\tilde\phi$ and $\mathcal F_P$ be its pulled-back amplitude and phase function, and let $\tilde\omega_{h^\vee}$ be the restriction to the fiber $\tilde Z_P^{h^\vee}$ of the pulled-back top form. For a continuous family of suitable integration contours $\tilde\Gamma=(\tilde\Gamma_{h^\vee})_{h^\vee\in\mathfrak h_{\mathbb R}^\vee}$ such that the integral converges, define

$$\tilde S_{\tilde\Gamma}(h^\vee,h)=\int_{\tilde\Gamma_{h^\vee}}e^{\mathcal F_P/\hbar}\tilde\phi(\,h)\,\tilde\omega_{h^\vee}.$$

Equivariant mirror conjecture. A full set of solutions to the $T$-equivariant quantum differential equations of $G/P$ is given by these integrals. This is the $T$-equivariant extension of the preceding mirror conjecture, with the covering recording the additional equivariant data; the paper states it as a conjecture and verifies the mirror construction only in the special case of $SL_n/B$.