Narayanan–Schacht conjecture on the threshold for non-linear Hamilton cycles

Let $G_r(n,p)$ be the random $r$-uniform hypergraph, and let $C^{(r)}_{n,\ell}$ be the $r$-uniform $\ell$-cycle on $n$ vertices. Write $Z(C^{(r)}_{n,\ell})$ for the number of copies of $C^{(r)}_{n,\ell}$ in $G_r(n,p)$. For integers $r>\ell>1$, assume $(r-\ell)\mid n$. Narayanan–Schacht conjecture. If $p=p(n)$ satisfies

$$\mathbb{E}[Z(C^{(r)}_{n,\ell})]\longrightarrow\infty,$$

then

$$\Pr\bigl[C^{(r)}_{n,\ell}\subset G_r(n,p)\bigr]\longrightarrow 1.$$

Narayanan and Schacht proved that the first-moment threshold is sharp, but above that threshold the expected number of cycles can already be exponentially large. The conjecture asks whether divergence of the expectation alone suffices for the appearance of a non-linear Hamilton cycle with high probability.