EPPA for n-hypertournaments

Determine whether the class of all finite n-hypertournaments has coherent EPPA, and determine the same for H4-free and O4-free 3-hypertournaments.

Background

The question is motivated by negative results for a different hypertournament notion and by the unresolved EPPA status of several reduct-like and higher-arity classes.

References

Does the class of all finite $n$-hy-per-tour-na-ments (see Section~\ref{sec:hypertournaments} for definitions) have (coherent) EPPA? What about the $H_4$-free 3-hypertour-naments or the $O_4$-free 3-hypertournaments?

— Twenty years of Nešetřil's classification programme of Ramsey classes  (2501.17293 - Hubička et al., 28 Jan 2025) in Question, subsection EPPA