Universality conjecture for graphs with bounded density

For $d \in \mathbb{Q}$ and $n \in \mathbb{N}$, let $\mathcal{H}_d(n)$ be the family of all graphs with $n$ vertices and density at most $d$, where the density of a graph $H$ is

$$m(H)=\max_{H'\subseteq H}\frac{e(H')}{v(H')}.$$

A graph is $\mathcal{H}_d(n)$-universal if it contains every graph in $\mathcal{H}_d(n)$ as a subgraph. Bounded-density universality conjecture. For every $d \in \mathbb{Q}$ with $d>1$ and every $n \in \mathbb{N}$, there exists a graph $G$ with

$$e(G)\leq C n^{2-1/d}$$

edges which is $\mathcal{H}_d(n)$-universal, where $C=C(d)$. This conjecture would unify the known and almost-optimal universality bounds for several families whose properties bound their density; the general bounded-density case remains open.