Converse of Theorem 3.2 (amenable-trace characterization)

Prove the converse of Theorem 3.2: For any imitation game G=(X,Y,A,B,λ) and perfect non-signalling correlation p∈C_ns(G), if p∈C_qa(G), then there exist a von Neumann algebra U, a *-homomorphism ρ: C⟨X,A⟩⊗_alg C⟨Y,B⟩→U, and an amenable tracial state τ on U such that τ∘ρ(e_a^x⊗f_b^y)=p(a,b|x,y) for all x∈X, y∈Y, a∈A, b∈B.

Background

Theorem 3.2 supplies a sufficient condition: existence of a von Neumann algebra U with an amenable trace τ and a *-homomorphism ρ inducing p implies p∈C_qa(G). The authors conjecture the converse direction, aiming for a necessary-and-sufficient characterization mirroring results known in related settings.

They obtain the converse in a special case (Theorem 3.1) under |X|=|A|=2 and additional injectivity assumptions on the GNS representation components, but the general proof remains open.