Ramsey expansion of H4-free 3-hypertournaments

Determine whether the class of all finite H4-free 3-hypertournaments has a precompact Ramsey expansion, and, if so, construct one with the expansion property, preferably in a finite language; if not, construct a non-trivial optimal Ramsey expansion. Extend the determination to H_{n+1}-free n-hypertournaments for n≥4.

Background

H4-free 3-hypertournaments are strong amalgamation classes in finite relational languages and can be viewed as higher-order analogues of linear orders. They are the only 3-hypertournament class in the cited classification for which the authors had not found a Ramsey expansion.

References

Does the class of all finite $H_4$-free 3-hy-per-tour-na-ments have a precompact Ramsey expansion? If yes, find one with the expansion property (is it in a finite language?). If not, can one still find a non-trivial Ramsey expansion which would be in some sense optimal (cf.)? What about $H_{n+1}$-free $n$-hypertour-na-ments for $n\geq 4$?

— Twenty years of Nešetřil's classification programme of Ramsey classes  (2501.17293 - Hubička et al., 28 Jan 2025) in Question 1, Section 1?; Section Open problems, subsection H4-free 3-hypertournaments