Off-diagonal ordered matching versus triangle

Determine whether there exists ε>0 such that every ordered matching M^< on n vertices satisfies R_<(M^<,K^<_3)≤O(n^{2-ε}).

Background

The general upper bound for an ordered matching versus an ordered triangle is O(n2/log n), while a construction gives a substantially smaller lower bound. The problem asks whether sparsity and ordering force a uniform power improvement over the quadratic scale.

References

Does there exist an $\varepsilon > 0$ such that every ordered matching $M<$ on $n$ vertices satisfies $R_<(M<,K<_3) \leq O(n{2-\varepsilon})$?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem prob-oderRam-matchingK3, Section 2.2

They constructed an ordered matching $M<$ on $n$ vertices satisfying R_<(M<,K<_3) \geq O\left(\left(\frac{n}{\log{n}\right){4/3}\right) and posed the following problem.

Does there exist an $\varepsilon > 0$ such that every ordered matching $M<$ on $n$ vertices satisfies $R_<(M<,K<_3) \leq O(n{2-\varepsilon})$?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Section 2, subsection “Off-diagonal Ordered Ramsey Numbers,” Problem 2 cited as [clfs17]