Generalized Ghys conjecture for ladder types

Determine whether the connected components of the subgraph of the veering ancestry graph consisting of all finite edge-orientable veering triangulations are in bijective correspondence with the tail-equivalence classes of even ladder types.

Background

The paper observes that drilling preserves restrictions on ladder types and introduces even ladder types and tail equivalence. It then asks whether these data completely classify the connected components of the edge-orientable subgraph of the ancestry graph.

References

Let $A+$ be the subgraph of $A$ with vertices corresponding to all finite edge-orientable veering triangulations and edges corresponding to all possible single drillings between them. Are connected components of $A+$ in a bijective correspondence with the tail equivalence classes of even ladder types?

— Drilling veering triangulations with applications to pseudo-Anosov flows  (2610.12302 - Parlak et al., 8 Oct 2026) in Section 5.4, Question 5.13 (label question:ghys:generalized)