Random subgraph Hamiltonicity conjecture for Cayley graphs

Let $G$ be a Cayley graph on $n$ vertices with degree $d$, and let $G_p$ be the random graph obtained by retaining each edge of $G$ independently with probability $p$. Random Cayley-subgraph Hamiltonicity conjecture. There exists a sufficiently large constant $C$ such that, whenever

$$p\geq C\log n/d,$$

with high probability $G_p$ has a Hamilton cycle. This proposed conjecture simultaneously generalises the Lovász conjecture and the Hamiltonicity threshold for random graphs; it is stated as a future direction and remains open.