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

Let $G/P$ be a partial flag variety, let $Z_P^{\mathfrak h^\vee}$ be the pullback of the mirror family to $(\mathfrak h^\vee)^{W_P}$, and let $\omega$ and $\mathcal F_P$ be respectively the pulled-back top form and phase function. The mirror datum is $(Z_P^{\mathfrak h^\vee},\omega,\mathcal F_P)$. Mirror conjecture. The integrals

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

defined from this mirror datum give solutions to the quantum differential equations of $G/P$. This conjecture asserts that oscillatory integrals on the Lie-theoretic mirror family recover the quantum differential system of the flag variety; the paper verifies the corresponding mirror conjectures in the special case of $SL_n/B$, while the general case is presented as a conjectural statement.