Vertical-surgery characterization of decreasing drillings

Determine whether every veering triangulation V′ that admits a veering parent with fewer tetrahedra can be obtained by vertical surgery from a veering triangulation V with fewer tetrahedra than V′.

Background

The paper gives examples in which drilling produces a veering parent with fewer tetrahedra and discusses a conjectural relationship between such drillings and vertical surgery. The authors explicitly note that this proposed explanation is only conjectural and ask whether all tetrahedron-decreasing cases arise from vertical surgery.

References

Suppose that $V'$ admits a veering parent with fewer tetrahedra than $V'$. Does it follow that $V'$ can be obtained by vertical surgery from a veering triangulation $V$ with fewer tetrahedra than $V'$?

— Drilling veering triangulations with applications to pseudo-Anosov flows  (2610.12302 - Parlak et al., 8 Oct 2026) in Section 6.3, Question 6.3 (subsection Decreasing the number of tetrahedra; no label provided)