Equality of homomorphism and blowup thresholds

Prove that the homomorphism threshold and the blowup threshold are equal for every graph H, namely, establish that δ_hom(H)=δ_B(H).

Background

The paper defines the homomorphism threshold δ_hom(H) as the minimum asymptotic minimum-degree proportion forcing every H-free graph to admit a homomorphism to a bounded-size H-free graph. It defines the blowup threshold δ_B(H) as the corresponding minimum-degree proportion forcing every maximal H-free graph to be a blowup of a bounded-size graph.

The inequalities δ_hom(H)≤δ_B(H) are immediate from the definitions, and the paper proves equality for cliques and for odd cycles in the blowup-threshold result. Whether equality holds for all forbidden graphs is left unresolved and is proposed as a general conjecture.