Minimality of alternating paths

Determine whether, for some positive integer n, an ordering P^<_n of the n-vertex path satisfies R_<(P^<_n)<R_<(AP^<_n).

Background

Computational experiments suggest that alternating paths may minimize ordered Ramsey numbers among all path orderings. The problem asks whether this apparent extremality fails for at least one path size and ordering.

References

For some positive integer $n$, is there an ordering $P<_n$ of the path $P_n$ on $n$ vertices such that $R_<(P<_n) < R_<(AP<_n)$?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem prob-ordRam-altMin, Section 2.4.2

The precise multiplicative factor in $R_<(P<_n)$ is unknown. Computer experiments by Balko, Cibulka, Kr{a}l, and Kyn\v{c}l indicate that ordered Ramsey numbers of alternating paths might be minimal among all path orderings, making it a particularly interesting class of ordered paths.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem~\ref{prob-ordRam-altMin}, Subsubsection “Alternating paths”