Odd eventual pp-increasing-rank

Construct a graph whose eventual pp-increasing-rank is an odd integer, or prove that the eventual pp-increasing-rank of every graph is even.

Background

The paper exhibits a graph with pp-increasing-rank 3, showing that the ordinary pp-increasing-rank need not be even, despite the identity between the relevant even and odd push-pop bounds. It remains unresolved whether an analogous example exists for the eventual pp-increasing-rank.

References

We don't know whether such example exists for the eventual pp-increasing rank:

— Graphs with Bounded Push-Pop Walk Orderings  (2610.19724 - Amarilli et al., 14 Oct 2026) in Appendix, Section “Open Problems,” subsection “Odd Ranks”