Alternating paths and minimum path orderings

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

Background

Computer experiments suggest that alternating paths may minimize ordered Ramsey numbers among all path orderings. The problem asks whether an ordering can have a strictly smaller ordered Ramsey number than the alternating path.

References

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.

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 Section 2, subsection “Specific Classes of Ordered Graphs,” subsection “Alternating paths,” Problem cited as [bckk13]