Big Ramsey degrees for the F-free 3-uniform hypergraph

Determine whether the universal countable 3-uniform hypergraph omitting a fixed four-vertex hypergraph with all but one possible edges has finite big Ramsey degrees and a big Ramsey structure.

Background

This is identified as a striking unresolved problem for a free amalgamation class in a finite language with a ternary relation and a finite forbidden substructure.

References

Let $\str{F}$ be a 3-uniform hypergraph on 4 vertices with all edges but one. Does the universal countable $\str{F}$-free 3-uniform hypergraph have finite big Ramsey degrees and a big Ramsey structure?

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