The boundary structure conjecture for special Lagrangian moduli spaces

Let $X$ be a compact Kähler manifold, let $D$ be an anticanonical divisor, let $U$ be a neighborhood of $D$, and let $M$ be the moduli space of pairs $(L,\nabla)$ consisting of a special Lagrangian submanifold $L$ of $(X\setminus D,\Omega)$ and a flat $U(1)$ connection on the trivial bundle over $L$. The boundary structure conjecture. Near its boundary, $M$ consists of pairs $(L,\nabla)$ such that $L\subset U\cap(X\setminus D)$ is a circle bundle over a special Lagrangian submanifold of $D$, and every fiber bounds a holomorphic disc of Maslov index $2$ contained in $U$. The conjecture is supported by the toric and non-toric examples discussed in the paper, but the asserted boundary description is not established in general.