Quantum-connection identification for the Lagrangian Grassmannian Landau–Ginzburg model

Let $X=LG(m)$, let $G_0^{W_t}$ be the Gauss–Manin module associated to $W_t$, and let $\bar G_0^{W_t}$ be the $\mathbb{C}[\hbar,e^t]$-submodule generated by the classes $[p_\lambda\omega]$, where $\lambda$ runs through the strict partitions inside an $m\times m$ box. Let ${}^A\nabla$ and ${}^B\nabla$ be the $A$- and $B$-model connections defined in the paper.

Quantum-connection identification conjecture. The differential operators $\hbar\,{}^B\nabla_{\partial_t}$ and $\hbar\,{}^B\nabla_{\hbar\partial_\hbar}$ preserve $\bar G_0^{W_t}$. Moreover, the assignment

$$\sigma^\lambda\longmapsto[p_\lambda\omega]$$

defines an isomorphism $H^*(X,\mathbb{C}[\hbar,e^t])\cong\bar G_0^{W_t}$ under which ${}^A\nabla$ is identified with ${}^B\nabla$.

This is the precise form of the paper's mirror-symmetry claim: the quantum connection of $LG(m)$ is recovered from the Gauss–Manin connection of $W_t$. The supplied text gives no resolution status.