Robust Bollobás–Eldridge–Catlin universality conjecture

For $2\leq k$, let $p^*_{k+1}$ denote the threshold function for the appearance of a $K_{k+1}$-factor in a random graph, and let $G_p$ be the random subgraph obtained by retaining each edge of $G$ independently with probability $p$. Let $k$-universality mean containing every graph on at most $n$ vertices with maximum degree at most $k$.

Robust universality conjecture. For any $k\geq 2$, there exists a constant $C>0$ such that, for all $n\in\mathbb N$ and $p\geq Cp^*_{k+1}$, every graph $G$ with $\delta(G)\geq \big(\tfrac{k}{k+1}\big)n$ satisfies that $G_p$ is $k$-universal with high probability.

This is presented as a common strengthening of the Bollobás–Eldridge–Catlin conjecture and the random-graph universality threshold. The parser marks it resolved, although the surrounding text describes it as a conjectural robustness statement; the claimed resolution should be checked against the cited source context.