The equality of Picard–Fuchs and Lagrangian Floer superpotentials

Let $X$ be a toric manifold with $-K_X$ numerically effective. Let $W^{\textrm{PF}}$ and $W^{\textrm{LF}}$ be the Picard–Fuchs and Lagrangian Floer superpotentials in the mirror, respectively, as introduced above. Superpotential equality conjecture.

$$W^{\textrm{PF}}=W^{\textrm{LF}}.$$

This conjecture asserts that the superpotential obtained from the Picard–Fuchs system agrees with the one defined using open Gromov–Witten invariants, providing a direct identification of the two constructions of the mirror for semi-Fano toric manifolds. The preceding results show that both superpotentials yield the quantum cohomology through their Jacobian rings, but their equality is proposed here rather than established in the supplied context.