Typical ordered matching versus triangle

Prove or refute that, for every positive integer χ, there exists ε(χ)>0 such that almost every n-vertex ordered matching with interval chromatic number χ has R_<(M^<,K^<_3)≤O(n^{2-ε(χ)}).

Background

This conjecture is a weaker typical-case version of the uniform subquadratic question for ordered matchings versus an ordered triangle. It concerns almost every ordered matching with a prescribed interval chromatic number.

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, Section 2.2