Equivalence of bounded merge-width and bounded flip-width

Prove or disprove that every graph class of bounded flip-width has bounded merge-width, and hence that graph classes of bounded merge-width and bounded flip-width coincide.

Background

The paper proves that bounded merge-width implies bounded flip-width. The converse would show that the decomposition-based merge-width framework captures precisely the classes defined by flip-width games, potentially yielding both conceptual simplification and algorithmic consequences such as approximation procedures.

References

We conjecture that the converse to \Cref{thm:mw-fw} also holds: that every class of bounded flip-width has bounded merge-width (see Discussion below).

— Merge-width and First-Order Model Checking  (2502.18065 - Dreier et al., 25 Feb 2025) in Section 1, paragraph following Corollary 2 and Section “Merge-width and flip-width”