The Barto–Opršal–Pinsker tractability dichotomy conjecture for reducts of finitely bounded homogeneous structures

Let $\rel B$ be a reduct of a finitely bounded homogeneous structure. A uniformly continuous minion homomorphism is a map from $\operatorname{Pol}(\rel B)$ to $\operatorname{Proj}$ preserving the minion operations and uniformly continuous with respect to the natural topology.

Barto–Opršal–Pinsker dichotomy conjecture. Exactly one of the following holds:

1. There exists a uniformly continuous minion homomorphism from $\operatorname{Pol}(\rel B)$ to $\operatorname{Proj}$, and $\operatorname{CSP}(\rel B)$ is $\textsf{NP}$-complete.
2. $\operatorname{Pol}(\rel B)$ does not have a uniformly continuous minion homomorphism to $\operatorname{Proj}$, and $\operatorname{CSP}(\rel B)$ is in $\textsf{P}$.

This is a formulation of the infinite-domain tractability dichotomy conjecture for first-order reducts of finitely bounded homogeneous structures. It has been verified in numerous special cases, but the general dichotomy remains open.