Extremal homomorphism bound for graphs with fixed minimum degree

Fix an integer $\delta\geq 1$ and a graph $H$. Let $\mathcal{G}(n,\delta)$ be the family of all $n$-vertex graphs with minimum degree $\delta$, and let $c(\delta,H)$ be a constant depending on $\delta$ and $H$. Minimum-degree homomorphism conjecture. For $n\geq c(\delta,H)$ and $G\in\mathcal{G}(n,\delta)$,

$$\hom(G,H) \leq \max\{\hom(K_{\delta+1},H)^{\frac{n}{\delta+1}}, \hom(K_{\delta,\delta},H)^{\frac{n}{2\delta}}, \hom(K_{\delta,n-\delta},H)\}.$$

This conjecture proposes three possible asymptotic extremal constructions: disjoint unions of $(\delta+1)$-cliques, disjoint unions of copies of $K_{\delta,\delta}$, and the complete bipartite graph $K_{\delta,n-\delta}$. It is motivated by exact results for independent-set colorings and remains open in the stated generality.