Sparse random graph vertex-minor universality conjecture

Let $p=\omega(1/\sqrt n)$ with $p\leq 1/2$, and let $G$ be sampled from either $\mathbb{G}(n,p)$ or $\mathbb{G}(n,1-p)$. A graph is $k$-vertex-minor universal if every graph on any specified set of $k$ vertices can be obtained as a vertex-minor. Sparse random graph universality conjecture. With high probability, $G$ is $k$-vertex-minor universal for some $k=\Omega(p\sqrt n)$. The conjecture extends the paper's random-graph universality methods beyond $p=1/2$. The stated range is open because the required argument must address correlations in the random walk arising when previously unrevealed edges are flipped.