Chi-boundedness of bounded flip-width classes

Determine whether every graph class of bounded flip-width is chi-bounded.

Background

Chi-boundedness asks whether chromatic number is bounded as a function of clique number throughout a graph class. The property is known for bounded twin-width and structurally bounded expansion classes, but its status for bounded flip-width remains unresolved.

References

It is unknown whether classes of bounded flip-width are $\chi$-bounded.

— Graph classes through the lens of logic  (2501.04166 - Pilipczuk, 7 Jan 2025) in Section 6, paragraph “Flip-width and dense analogues of sparse notions”