The tractability conjecture for CSPs of reducts of finitely bounded homogeneous structures

Let $\mathfrak{B}$ be a normal representation of a finite relation algebra $\mathbf{A}$. A primitive-positive interpretation is an interpretation defined by primitive-positive formulas, and $\mathfrak{B}$ pp-constructs a structure if it pp-interprets a homomorphically equivalent structure. Let $(\{0,1\};\operatorname{NAE})$ denote the Boolean structure with the not-all-equal relation. Write $\operatorname{CSP}(\mathfrak{B})$ for the constraint satisfaction problem of $\mathfrak{B}$ and $\operatorname{NSP}(\mathbf{A})$ for the network satisfaction problem of $\mathbf{A}$.

Tractability conjecture. If $\mathfrak{B}$ does not pp-construct $(\{0,1\};\operatorname{NAE})$, then

$$\operatorname{CSP}(\mathfrak{B})\text{ and }\operatorname{NSP}(\mathbf{A})\text{ are in }\operatorname{\mathbf{P}}.$$

If the stated pp-construction condition holds, the CSP is known to be $\operatorname{\mathbf{NP}}$-hard; the conjecture asserts polynomial-time tractability in the complementary case. It is presented as a specialization of the broader dichotomy conjecture for reducts of finitely bounded homogeneous structures.