Ordered paths with interval chromatic number 2

Determine whether every ordering P^< of the n-vertex path with interval chromatic number 2 satisfies R_<(P^<)≤O(n^2).

Background

For such ordered paths, the survey records an O(n3) upper bound and an Ω((n/log n)2) lower bound. The question asks whether the upper bound can be reduced to quadratic order.

References

Is it true that $R_<(P<) \leq O(n2)$ for every ordering $P<$ of the path on $n$ vertices with interval chromatic number 2?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following discussion of Theorem (thm-oderRam-degreeIntNum2), Section 2