Hamiltonian cycle game's threshold conjecture

Let ${\cal H}$ be the set of Hamiltonian cycles in $K_n$. For a family of winning sets ${\cal F}$, let $b_{\cal F}^p$ denote the smallest bias for which Breaker wins the random game on the edges of $G(n,p)$, and let $p_{\cal F}$ denote the threshold probability for Maker's win. Hamiltonian cycle game's threshold conjecture. There exists a constant $C$ such that

$$b_{{\cal H}}^p = \Theta\left(p\frac{n}{\log n}\right), \qquad p\geq C\frac{\log n}{n}.$$

In particular, $p_{{\cal H}}=\frac{\log n}{n}$. This conjecture asserts that the Hamiltonian cycle game has the same threshold behaviour as the connectivity and perfect matching games; the paper establishes corresponding results for several other games but leaves this Hamiltonian-cycle assertion open.