Precompact Ramsey expansion of finite groups

Determine whether the class of all finite groups has a precompact Ramsey expansion.

Background

Finite groups form an amalgamation class and have EPPA, making them a natural test case for the conjectured relationship between strong amalgamation, EPPA, and Ramsey expansions. The survey notes that the authors weakly expect a negative answer.

References

Does the class of all finite groups have a precompact Ramsey expansion?

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