Let me go from the weakest to strongest sense in which the conjecture should
First, at the purely topological level, it is true for any Calabi-Yau variety
with a toric degeneration whose dual intersection complex is ``simple''.
These notions are part of my program with Bernd Siebert: see our paper
http://arxiv.org/abs/math/0309070 for the definitions of toric degeneration
and simple. In http://arxiv.org/abs/math/0406171 I proved that all
Calabi-Yau varieties arising in the Batyrev-Borisov construction as
complete intersections in toric varieties have such degenerations.
The problem is that Bernd and I have been putting off writing the paper linking
the logarithmic approach to topological fibrations for years now, largely
due to lack of motivation. So there is no reference in the literature yet
for this result. I do hope we will finally complete this paper next year.
Second, at the Lagrangian level, there are the results of W.-D. Ruan you
mentioned. In addition, Castano-Bernard and Matessi in
http://arxiv.org/abs/math/0611139 showed that given an affine three-manifold $B$
with ``simple'' singularities, one can construct a symplectic six-manifold
along with a Lagrangian fibration to $B$. So one can apply this to the case
where $B$ is the intersection complex of a polarized toric degeneration of
Calabi-Yau threefolds. One expects this six-manifold to be symplectomorphic
to a general fibre of the degeneration, but there is no proof of this at the
Finally, at the special Lagrangian level, I think it is safe to say there
are no known examples on compact non-singular Calabi-Yau threefolds with
non-degenerate metric. There are some examples for non-compact
Calabi-Yau varieties, specifically toric ones, see my paper