Kohayakawa–Nagle–Parczyk conjecture on squares of Hamilton cycles in random graphs

Let $G(n,p)$ be the binomial random graph on vertex set $[n]$, with each possible edge included independently with probability $p=p(n)$. A square of a Hamilton cycle is the second power of a Hamilton cycle: for some cyclic ordering of the vertices, every pair of vertices at cyclic distance at most two is an edge. Kohayakawa–Nagle–Parczyk conjecture. For every fixed $\varepsilon>0$, if

$$p\geq (1+\varepsilon)\sqrt{e/n},$$

then $G(n,p)$ a.a.s. contains a square of a Hamilton cycle. This is a more precise proposed sufficient condition than the earlier threshold prediction; the supplied passage does not state whether it has been resolved.