Finite big Ramsey degrees for Fraïssé limits with finite constraints

Let $\mathcal{K}$ be a relational Fraïssé class with finitely many relations and a finite constraint set. A constraint set is a finite set of finite structures such that membership in $\mathcal{K}$ is characterized by omitting these structures as induced substructures. Suppose that $\mathcal{K}$ satisfies the Ramsey property or has finite small Ramsey degrees. Finite big Ramsey degrees conjecture. The Fraïssé limit of $\mathcal{K}$ has finite big Ramsey degrees.

The conjecture proposes that finite small Ramsey degrees, together with finite relational complexity and finitely many constraints, suffice to obtain finite big Ramsey degrees for the corresponding homogeneous structure. The suggested approach is to generalize strong coding-tree and forcing methods from Henson graphs and the Rado graph to Fraïssé limits in arbitrary finite relational languages; its resolution is not supplied here.