Bullet-boundedness of arbitrary symmetric forbidden-colouring families

Prove or disprove that every symmetric family X of s-edge-colourings of K_k is bullet-bounded, meaning that the set of basic optimal solutions to the loop-extended optimization problem Q^\bullet(X) is nonempty and has a uniformly bounded number of parts.

Background

The paper introduces Q\bullet(X), an extension of its optimization framework allowing colours on edges within parts, in order to capture cases such as the three-colour rainbow triangle problem that are not bounded in the original framework. Bullet-boundedness is needed to obtain a finite structural optimization problem and is characterized as a Ramsey-type property. The authors establish it for the three-colour rainbow triangle but leave the general case open.

References

For all integers $s \geq 2$ and $k \geq 3$ and every (symmetric) family $X$ of $s$-edge colourings of $K_k$, is $X$ $\bullet$-bounded?

— A framework for the generalised Erdős-Rothschild problem and a resolution of the dichromatic triangle case  (2502.12291 - Gupta et al., 17 Feb 2025) in Problem statement in Section 6.1, 'A more general framework'