The stable-labeling variational asymptotic conjecture

Fix an $r$-graph $H$ with maximum degree $\Delta$, and let $\rho_H(\delta)$ be the infimum of the volumes of compatible collections of mixed hubs that respect stable labelings and satisfy the source's condition $P_H(\mathcal{S},c)\ge 1+\delta$. Let $\mathcal{G}_n$ be the class of edge-weighted $r$-graphs on $[n]$, let $I_p$ be the variational cost, and let $t(H,G_n)$ be the homomorphism density.

Stable-labeling variational asymptotic conjecture. If $n\to\infty$ and $p=p(n)$ satisfies $n^{-1/\Delta}\ll p\ll 1$, then

$$\min\{I_p(G_n):G_n\in\mathcal{G}_n,\ t(H,G_n)\ge (1+\delta)p^{|E(H)|}\}\sim \frac{1}{r!}n^r p^\Delta\rho_H(\delta)\log(1/p).$$

This is the formal asymptotic version of the preceding stable-labeling mixed-hub proposal. The supplied text does not provide a proof or resolution.