Polynomial equivalence with Boolean CSPs for complete coloured patterns

Determine whether, for every finite set F of 2-edge-coloured complete graphs, the F-free 2-edge-colouring problem and the Boolean constraint satisfaction problem CSP(B_F) are polynomial-time equivalent.

Background

The paper establishes a general polynomial-time reduction from the F-free 2-edge-colouring problem to a Boolean CSP with template B_F. For sets consisting of coloured triangles, the authors prove polynomial-time equivalence. They ask whether this equivalence extends to every finite set of 2-edge-coloured complete graphs; a positive answer would imply a P-versus-NP-complete dichotomy for these colouring problems.