The special Lagrangian fibration conjecture for Calabi–Yau double covers

Let $(X,\omega,J)$ be a compact Kähler manifold, let $H$ represent twice the anticanonical class, and let $Y$ be the Calabi–Yau double cover of $X$ branched along $H$. Let $\bar B$ be a singular affine manifold with boundary and two affine structures. Calabi–Yau double-cover fibration conjecture. The following should hold: $X$ carries a special Lagrangian fibration (or foliation) $f:X\to\bar B$ whose generic fibers are special Lagrangian tori in $X\setminus H$ and whose fibers over $\partial\bar B$ are special Lagrangians with boundary in $H$; and $Y$ carries a special Lagrangian torus fibration $\tilde f:Y\to\tilde B$, where $\tilde B$ is a singular affine manifold without boundary and with two affine structures, obtained by gluing two copies of $\bar B$ along their boundary. This is the precise form of the paper's main conjecture. The source emphasizes that no general theorem currently produces the required fibration from the one on $X\setminus H$, particularly in the presence of singular fibers.