The homomorphic-equivalence CSP dichotomy conjecture

Let $\mathbb{A}$ be a reduct of a finitely bounded homogeneous structure. Let $\mathbb{S}$ be the finite structure defined by

$$\mathbb{S}:=(\{0,1\};\{(0,0,1),(0,1,0),(1,0,0)\}).$$

A structure is homomorphically equivalent to another when homomorphisms exist in both directions, and a primitive-positive interpretation is an interpretation defined by primitive-positive formulas.

Homomorphic-equivalence CSP dichotomy conjecture. One of the following holds:

1. $\mathbb{S}$ is homomorphically equivalent to a structure with a pp-interpretation in $\mathbb{A}$, and consequently $\operatorname{CSP}(\mathbb{A})$ is NP-complete.
2. $\operatorname{CSP}(\mathbb{A})$ is polynomial-time solvable.

This is the younger dichotomy conjecture described in the source. It seeks to avoid passing first to a model-complete core; the paper proves that the two conjectures are equivalent, while the general dichotomy remains unresolved.