The boundary-behavior conjecture for special Lagrangian fibrations

Let $X$ be a Kähler manifold, let $D\subset X$ be a smooth Calabi–Yau hypersurface in the anticanonical linear system, and let $\pi:X\setminus D\to B$ be a reasonable special Lagrangian fibration, assuming that the Kähler metric on $X$ is complete. Let $\partial B$ denote the boundary of the base. Boundary-behavior conjecture. Near $\partial B$, the fibers of $\pi$ should be contained in a neighborhood of $D$, and the smooth fibers should be $S^1$-bundles over special Lagrangian tori in $(D,\omega_{|D},\Omega_D)$. This conjectural behavior describes how the fibration on the complement of an anticanonical divisor should approach the divisor and motivates the relative mirror-symmetry constructions in the paper.