The pp-construction conjecture for hard finite-domain PCSPs over homogeneous reducts

Let $G$ and $H$ be finite graphs, and let $S$ be a countably infinite graph that is a reduct of a finitely bounded homogeneous structure. Suppose that

$$G\to S\to H.$$

A primitive-positive, or pp, construction interprets one relational structure in another using primitive-positive formulas.

Pp-construction conjecture. If $\operatorname{PCSP}(G,H)$ is NP-hard, then $K_3$ has a pp-construction in $S$.

This proposes a structural explanation for NP-hardness in the indicated sandwich setting. The paper explicitly leaves the assertion as an open problem; it also notes that, if P differs from NP, the preceding tractability conjecture would imply it.