[Update]

Margalit has been stating a more precise version of this conjecture since at least 2012. See page 38 of his slides.

[Old version]

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

[Update]

Margalit has been stating a more precise version of this conjecture since at least 2012. See page 38 of these slides.

[Old version]

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

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

[Update]

Margalit has been stating a more precise version of this conjecture since at least 2012. See page 38 of his slides.

[Old version]

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