There is related work by several people.  I could not find a statement of the conjecture, but I have heard versions of it - sometimes it is generalized to more than just the minimal dilatation pA map. Instead, there is a statement about all pA maps satisfying certain bounds.  Here are a few relevant references.

* Small dilatation pseudo-Anosovs and 3–manifolds - by Benson Farb, Christopher J. Leininger, and Dan Margalit.
* Ideal triangulations of pseudo-Anosov mapping tori - by Ian Agol.
* Penner sequences and asymptotics of minimum dilatations for subfamilies of the mapping class group - by Eriko Hironaka