Finiteness for pseudo-Anosov flows without negative lozenges

Let $M$ be a closed $3$-manifold. A pseudo-Anosov flow without negative lozenges is a pseudo-Anosov flow having no negative lozenges. Finiteness conjecture. There are finitely many pseudo-Anosov flows without negative lozenges on $M$ up to orbit equivalence. This is a potentially approachable special case of the general Finiteness Conjecture; the paper presents it as an open question.