The naive mirror construction conjecture for anticanonical complements

Let $(X,\omega,J)$ be a compact Kähler manifold, let $D$ be an anticanonical divisor in $X$, and let $\Omega$ be a holomorphic volume form on $X\setminus D$. A complexified moduli space is the moduli space of special Lagrangian tori in $X\setminus D$ equipped with flat $U(1)$ connections. The naive mirror construction conjecture. A mirror manifold $M$ can be constructed as this moduli space, with a superpotential $W:M\to\mathbb{C}$ given by Fukaya–Oh–Ohta–Ono's $m_0$ obstruction to Floer homology. Moreover, the fiber of this Landau–Ginzburg model is mirror to $D$. This conjecture proposes a direct extension of the SYZ picture from Calabi–Yau manifolds to complements of anticanonical divisors and Fano geometry; the paper presents it as a naive formulation whose general validity remains open.