Typical ordered matchings versus triangles

Prove or refute that, for every positive integer chi, there exists epsilon(chi)>0 such that R_<(M^<,K^<_3) <= O(n^{2-epsilon(chi)}) for almost every ordered matching M^< on n vertices with interval chromatic number chi.

Background

The survey presents this as a weaker variant of the uniformly quantified matching problem. It concerns almost every ordered matching with a prescribed interval chromatic number and conjectures a power-saving over the general quadratic-scale upper bound.

References

For a positive integer $\chi$, there is a constant $\varepsilon(\chi)>0$ such that $R_<(M<,K<_3) \leq O(n{2-\varepsilon(\chi)})$ for almost every ordered matching $M<$ on $n$ vertices with interval chromatic number~$\chi$.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Conjecture following Problem prob-oderRam-matchingK3, subsection “Off-diagonal Ordered Ramsey Numbers”