The BAS characterization conjecture for pseudo-Anosov flows

Let $\phi$ be a transitive pseudo-Anosov flow on an oriented closed $3$-manifold. Let $\operatorname{sing}(\phi)$ denote its singular orbits, and let positive and negative lozenges be the corresponding configurations in its orbit space. A BAS characterization conjecture. If $\phi$ does not admit both positive and negative lozenges disjoint from $\operatorname{sing}(\phi)$, then $\phi$ is BAS. The paper notes that having both positive and negative lozenges disjoint from singular orbits obstructs BAS; the converse is conjectured and remains open.