Asymptotics for bounded-bandwidth ordered Ramsey numbers

Establish whether R_<(P^<_{k,n}) = n^{2+o(1)} as n tends to infinity.

Background

The best upper bound stated in the survey is n{4+o(1)}, while only a quadratic lower bound is known. The cited authors believe the correct growth is closer to the lower bound and formulate the displayed asymptotic question.

References

Gir~{a}o, Janzer, and Janzer believed that the truth is closer to the lower bound and posed the following problem.

Is $R_<(P<_{k,n})= n{2+o(1)}$ as $n \to \infty$?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Section 2, subsection “Bounded bandwidth,” Problem cited as [gjj24]

Is $R_<(P<_{k,n})= n{2+o(1)}$ as $n \to \infty$?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following Theorem (thm-ordRam-k), subsection “Bounded bandwidth”