Homomorphism and blowup threshold conjecture

Let $H$ be a graph. The homomorphism threshold $[?]$ and blowup threshold $[?]$ are defined for $H$ as the corresponding minimum-degree thresholds for forcing the relevant structural conclusion.

Homomorphism–blowup threshold conjecture. For any graph $H$,

$$\delta_{\textup{hom}}(H)=\delta_{\textup{B}}(H).$$

The paper establishes the inequality $\delta_{\chi}(H)\leq \delta_{\textup{hom}}(H)\leq \delta_{\textup{B}}(H)$ and proves the blowup threshold for odd cycles, but equality for every graph remains open.