Lower bounds for bounded-degree, interval-chromatic-two ordered graphs

Determine whether there exists a constant c>0 such that, for every fixed maximum degree Δ, some sequence of n-vertex Δ-regular ordered graphs with interval chromatic number 2 has ordered Ramsey number at least n^{cΔ}.

Background

Polynomial upper bounds are known for bounded-degree ordered graphs with bounded interval chromatic number, but the survey states that no nontrivial lower bounds are known in this setting. The problem asks whether substantially large polynomial ordered Ramsey numbers can occur even for interval chromatic number 2.

References

However, we have no non-trivial lower bounds for this case and thus Balko, Cibulka, Kr{a}l, and Kyn\v{c}l stated the following problem.

Is there a constant $c>0$ such that for every fixed $\Delta$ there is a sequence ${G<n}{n\in\mathbb{N}$ of ordered $\Delta$-regular graphs $G<_n$ with $n$ vertices and interval chromatic number $2$ such that $R_<(G<_n) \ge n{c\Delta}$?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following Equation (eq-ordRam-degInt), Section 2