Expectation-threshold conjecture for nonlinear Hamiltonian cycles

Let $G^{(r)}(n,p)$ be the binomial random $r$-uniform hypergraph, and let $X_\ell$ count its Hamiltonian $\ell$-cycles. Expectation-threshold conjecture. For all integers $r > \ell > 1$, if $p=p(n)$ satisfies

$$\mathbb{E}[X_\ell]\to\infty$$

as $n\to\infty$, then

$$\mathbb{P}\bigl(G^{(r)}(n,p)\text{ contains a Hamiltonian $\ell$-cycle}\bigr)\to 1.$$

The paper establishes the sharp threshold up to its asymptotic order and conjectures that the expectation threshold determines the critical window more precisely; the proposed statement remains open.