Strong amalgamation classes with EPPA have precompact Ramsey expansions

A class of finite structures has EPPA if every structure in the class embeds into another structure in the class such that every partial automorphism of the first extends to an automorphism of the second. A class has strong amalgamation when amalgamation introduces no identifications outside the common substructure, and a Ramsey expansion is precompact when each finite structure has only finitely many expansions up to isomorphism.

EPPA–Ramsey expansion conjecture. Every strong amalgamation class with EPPA has a precompact Ramsey expansion.

This conjecture is stated among the paper's open problems and connects extension properties for partial automorphisms with structural Ramsey theory. The source gives no general proof or resolution.