Connectivity of the edge-orientable ancestry graph

Determine whether any two edge-orientable veering triangulations with four ladders in each boundary component lie in the same connected component of the veering ancestry graph A.

Background

The paper proves that connected components of the veering ancestry graph correspond to almost orbit-equivalence classes of transitive pseudo-Anosov flows. Under this correspondence, Ghys’s conjecture becomes the stated connectivity problem for edge-orientable veering triangulations with four ladders per boundary component.

References

Any two edge-orientable veering triangulations with four ladders in each boundary component are in the same connected component of $A$.

— Drilling veering triangulations with applications to pseudo-Anosov flows  (2610.12302 - Parlak et al., 8 Oct 2026) in Section 5.4, Conjecture 5.12 (label conj:veering:graph:component)