EPPA for oriented graphs omitting large independent sets

Determine for which k≥2 there exists ℓ such that every finite oriented graph with no independent set of size k embeds into an oriented graph with no independent set of size ℓ that extends all partial automorphisms of the original graph.

Background

This generalizes the tournament case k=2. Existing constructions yield independent sets whose size grows with the size of the structure, so they do not answer the uniform-boundedness question.

References

For which $k\geq 2$ is there $\ell$ such that for every oriented graph $\str A$ which contains no independent set of size $k$ there is an oriented graph $\str B$ which contains no independent set of size $\ell$ such that $\str A\subseteq \str B$ and every partial automorphism of $\str A$ extends to an automorphism of $\str B$?

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