Product pp-constructibility for acyclic CSP reductions

Prove or disprove that, for every finite digraph H, the infinite product digraph H×(,<) pp-constructs H, where (,<) is the strict linear order used to encode acyclic instances.

Background

The paper shows that CSP(H) reduces to CSP(H) restricted to acyclic digraphs and that the latter problem is polynomial-time equivalent to CSP(H×(,<)). This raises an algebraic question about whether the product construction preserves enough expressive power to pp-construct H itself.

A positive answer would provide a structural explanation for the equivalence between ordinary and acyclic-instance complexity.