The Strominge-Yau-Zaslow conjecture is roughly the following. Any Calabi-Yau $m$-manifold $X$ admits a special Lagrangian $T^m$ fibration (maybe at around a special point in its complex moduli space) and a mirror partner $Y$ is obtained by dualizing the tori $T^m$ with "instanton corrections" coming from singular fibers.

If my memory serves one of heuristics of SYZ conjecture comes from Kontsevich's homological mirror symmetry conjecture; the moduli space of sky-scraper sheaves $\mathcal{O}_y$ (the easiest B-branes) on $Y$ is $Y$ itself, and there should be a corresponding moduli space of A-branes on $X$. A-brane is a pair $(L,c)$ of Lagrangian submnaifold $L\subset X$ and a flat $U(1)$ connection $c$ on $L$. By computing cohomology groups $HF^*(L,L)\cong Ext(\mathcal{O}_y,\mathcal{O}_y)=H^*(T^m,T^m)$, we expect that our $L$ is a Lagrangian $T^m$. My first question is

How do you expect that $L$ is a spcecial Lagrangian manifold?

If we assume that $L$ is special Lagrangian, then its deformation space is known and is of dimension $3$, and we expect that $L$ sweeps over $X$. My second question is

Does the flat $U(1)$ connection $c$ on $L$ play any role in this story?

All we "deduced" from HMS conjecture is that $X$ admits a special Lagrangian fibration. Can we say anything more? Since HMS conjecture doesn't say anything about construction of mirror manifolds, I am afraid that we cannot say anything about dualizing these $L=T^n$ etc. My third question is

What is the A-brane object on $X$ that corresponds to the obvious $6$-brane $Y$?