Polynomial-time homomorphism-embedding CSPs for homogeneous Ramsey structures

Let $\str M$ be a homogeneous Ramsey structure in a finite relational language. A homomorphism-embedding is a map that is a homomorphism and restricts to an embedding on every irreducible substructure; the corresponding homomorphism-embedding constraint satisfaction problem asks whether such a map exists from the finite input structure to the target.

Homomorphism-embedding CSP conjecture. There is a homogeneous Ramsey structure $\str H$ in a finite relational language, first-order bi-interpretable with $\str M$, such that the homomorphism-embedding variant of the constraint satisfaction problem of $\str H$ is solvable in polynomial time.

The conjecture formalizes the observation that known Ramsey classes often admit polynomial-time algorithms through completion methods. The source presents it as open.