Balogh–Morris–Samotij diameter-game threshold conjecture

Let $s\geq 3$ and let $\mathcal{D}_{s,n}$ be the $(1:b)$ Maker–Breaker game on the edges of $K_n$ in which Maker wins precisely when her spanning subgraph has diameter at most $s$. Write $b_{\mathcal{D}_{s,n}}$ for its threshold bias. Diameter-game threshold conjecture. The threshold bias $b_{\mathcal{D}_{s,n}}$ is close to the bound for Breaker's side, namely the bound in part (b) of the stated theorem, of order $n^{1-1/(s-1)}$. This conjecture concerns the asymptotic sharpness of the Breaker-side bound; the source does not specify an exact asymptotic equivalent or establish its resolution.