The random-graph conjecture for external and internal bisections

For a graph on $n$ vertices, a bisection is a red-blue colouring in which the colour-class sizes are equal, or differ by one when $n$ is odd. An external bisection has at least half of every vertex's neighbours in the opposite colour, while an internal bisection has at least half in its own colour. Let $p_n\in[0,1]$ be allowed to depend on $n$, and let $G\sim\mathbb{G}(n,p_n)$ be the Erdős–Rényi random graph. Random-graph bisection conjecture. For any $p_n\in[0,1]$, with high probability $G$ has an external bisection. If $(1-p_n)n-\log n\to\infty$, then with high probability $G$ also has an internal bisection. The external assertion is stated for all edge probabilities, while the internal assertion excludes the regime in which the graph is too close to complete; the source records this conjecture as open.