Toric mirror symmetry conjecture for the $\frac{\infty}{2}$ variation of Hodge structure

Let the toric data satisfy $\hat{\rho}\in \operatorname{cl}(\tilde{C}_\mathcal{X})$. Let $\mathcal{F}_{\rm B}$ be the B-model $\frac{\infty}{2}$ variation of Hodge structure associated with the Landau–Ginzburg mirror of $\mathcal{X}$, let $\mathcal{F}_{\rm A}=(\widetilde{\mathcal{F}}_{\rm A}\to U)/H^2(\mathcal{X},\mathbb Z)$ be the A-model $\frac{\infty}{2}$ variation, and let $\tau$ be the map to the A-model parameter space. Toric mirror symmetry conjecture. There exists an isomorphism of graded $\frac{\infty}{2}$ variations of Hodge structure

$$\operatorname{Mir}\colon \mathcal{F}_{\rm B}\cong \tau^*\mathcal{F}_{\rm A}$$

that sends the section $[e^{W_q/z}\omega_q]$ of $\mathcal{F}_{\rm B}$ to the $I$-function $I(q,z)\in\mathcal{H}^{\mathcal{X}}$, in the sense that

$$\mathcal{J}_{\tau(q)}(\operatorname{Mir}[e^{W_q/z}\omega_q])=I(q,z),$$

where $\mathcal{J}_{\tau(q)}\colon \widetilde{\mathcal{F}}_{{\rm A},\tau(q)}\to\mathcal{H}^{\mathcal{X}}$ is the embedding given by the fundamental solution. This conjecture is the mirror-symmetry identification between the Landau–Ginzburg B-model and the quantum-cohomological A-model for the toric orbifold; the supplied text gives no resolution status.