Ascoli–Frederickson–Frederickson–McFarland–Post vertex-minor universality conjecture

Let $G(n,p)$ denote the Erdős–Rényi random graph on $n$ vertices, and say that a graph $G$ is $k$-vertex-minor universal if, for every vertex subset $U\subseteq V(G)$ of size $k$, every graph on $U$ is a vertex-minor of $G$. Here, $G\sim G(n,p)$ means that each edge is present independently with probability $p$.

Vertex-minor universality conjecture. If

$$p=\omega(1/\sqrt{n}),\qquad p\leqslant 1/2,$$

and $G\sim G(n,p)$ or $G\sim G(n,1-p)$, then, with high probability, $G$ is $k$-vertex-minor universal for some

$$k=\Omega(p\sqrt{n}).$$

This conjecture extends the known result for $G(n,1/2)$, where almost all graphs are $O(\sqrt{n})$-vertex-minor universal, and predicts the corresponding scale for sparse and complementary random graphs. The source gives no resolution of the conjecture.