Finiteness conjecture for algebraically primitive Teichmüller curves in genus three

A Teichmüller curve in ${\mathcal M}_{3}$ is algebraically primitive when its trace field has degree $3$. Finiteness conjecture. There are only finitely many algebraically primitive Teichmüller curves in ${\mathcal M}_{3}$. Algebraically primitive Teichmüller curves are central to the classification of Teichmüller curves in genus three; the paper presents strong numerical and theoretical evidence, but does not establish finiteness.