Separation between two- and three-color off-diagonal numbers

Determine whether there exist n-vertex ordered matchings M^< for which the ratio R_<(K^<_3,K^<_3,M^<)/R_<(K^<_3,M^<) tends to infinity as n tends to infinity.

Background

This question asks whether adding a second ordered-triangle color can increase the off-diagonal ordered Ramsey number by an unbounded factor, paralleling a classical conjectural phenomenon for ordinary Ramsey numbers.

References

Are there ordered matchings $M<$ on $n$ vertices such that $$\lim_{n \to \infty} \frac{R_<(K<_3, K<_3, M<)}{R_<(K<_3, M<)} = \infty?$$

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following off-diagonal multicolor bounds, Section 2.5