Sharp threshold conjectures for Maker's matching and Hamiltonian cycle games

Let $G$ be the evolving random graph process, and write $\tau(\mathcal E)$ for the hitting time at which event $\mathcal E$ first occurs. Let ${\cal M}$ be the perfect matching game and ${\cal H}$ the Hamiltonian cycle game. Sharp threshold conjectures for Maker's matching and Hamiltonian cycle games.

$$\tau(\text{Maker wins }{\cal M})=\tau(\delta(G)\geq 2),$$

and

$$\tau(\text{Maker wins }{\cal H})=\tau(\delta(G)\geq 4).$$

These conjectures propose sharp hitting-time thresholds for Maker's wins in the perfect matching and Hamiltonian cycle games. The surrounding discussion notes that the connectivity game has an established sharp threshold, while these stronger assertions for the matching and Hamiltonian games are posed as open questions.