Random-graph square Hamilton-cycle conjecture

Let $G_{n,p}$ be the binomial random graph. The square of a Hamilton cycle is obtained from a Hamilton cycle by adding edges between all vertices at distance at most two on the cycle. Random-graph square Hamilton-cycle conjecture. If $p\gg n^{-1/2}$, then asymptotically almost surely $G_{n,p}$ contains the square of a Hamilton cycle. The source presents this as open and motivated by the expected first-moment threshold.