Bodirsky–Pinsker CSP dichotomy conjecture for reducts of finitely bounded homogeneous structures

Let $L$ be a finite relational language, let $\str M$ be an $L$-structure, and let a first-order reduct of a structure mean a structure whose relations are first-order definable in that structure. A homogeneous structure is one in which every isomorphism between finite substructures extends to an automorphism, and it is finitely bounded when its age is defined by forbidding finitely many finite structures. The constraint satisfaction problem $\operatorname{CSP}(\str M)$ asks, for a finite $L$-structure $\str A$, whether there is a homomorphism $\str A\to\str M$.

Bodirsky–Pinsker conjecture. If $\str M$ is a first-order reduct of a finitely bounded homogeneous structure in a finite relational language, then either $\operatorname{CSP}(\str M)$ is solvable in polynomial time or it is NP-complete.

This conjecture extends the finite CSP dichotomy theorem to the indicated class of infinite structures. The source says that it has been verified in several concrete instances, but remains unresolved in general.