Stojaković–Szabó threshold-bias conjecture for the random-graph Hamiltonicity game

Let $G\sim G(n,p)$ be a random graph on $n$ labeled vertices, where each pair of vertices is independently included as an edge with probability $p$. Let $\mathcal H(G)$ denote the Maker–Breaker Hamiltonicity game played on the edge set of $G$, and let $b^*$ be its threshold bias.

Stojaković–Szabó conjecture. There exists a constant $C$ such that for every

$$p\geq \frac{C\ln n}{n},$$

a random graph $G\sim G(n,p)$ is typically such that

$$b^*=\Theta\left(\frac{np}{\ln n}\right).$$

The conjecture concerns the asymptotic threshold bias for Hamiltonicity on random boards and remains unresolved in the supplied context.