Quadratic upper bound for interval-chromatic-number-two ordered paths

Prove that every ordering P^< of the n-vertex path with interval chromatic number 2 satisfies R_<(P^<) = O(n^2).

Background

For ordered paths with interval chromatic number 2, the survey gives an O(n3) upper bound and an Ω((n/log n)2) lower bound. The cited authors explicitly ask 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 Section 2, subsection “Bounded degrees and interval chromatic number”