Degree-threshold conjecture for linear Hamiltonian cycles

Let $G^{(r)}(n,p)$ be the binomial random $r$-uniform hypergraph. For $r\geq 3$, define

$$p^{\mathrm{deg}}_r(n)=\frac{(r-1)!\log n}{n^{r-1}}.$$

This is the sharp threshold for the disappearance of isolated vertices. Degree-threshold conjecture. For each $r\geq 3$, $p^{\mathrm{deg}}_r(n)$ is the sharp threshold for the appearance of a linear Hamiltonian cycle in $G^{(r)}(n,p)$. The paper proves that the appearance of a linear Hamiltonian cycle has a sharp threshold and predicts that it coincides with the isolated-vertex threshold; the coincidence remains open.