Universal bounded-degree graph conjecture

Let $\mathcal H(n,\Delta)$ be the family of all graphs on $n$ vertices with maximum degree $\Delta$. A graph $G$ is $\mathcal F$-universal if it contains every member of a graph family $\mathcal F$. Let $G_{n,p}$ be the binomial random graph.

Universal bounded-degree graph conjecture. Let $\Delta\in\mathbb N$ and $p=\omega\left(n^{-1}\log^{1/\Delta}n\right)^{\frac{2}{\Delta+1}}$. Then, with high probability, $G_{n,p}$ is $\mathcal H(n,\Delta)$-universal.

This strengthens the preceding embedding conjecture by requiring one random graph to contain every graph on $n$ vertices of maximum degree at most $\Delta$. The proposed density is motivated by the conjectured worst case of $K_{\Delta+1}$-factors, but the statement remains open in the source.