Regular hypergraph loose Hamilton cycle conjecture

Let $\mathbb H^{(k)}(n,d)$ denote the random $d$-regular $k$-uniform hypergraph on $n$ vertices, and let a loose Hamilton cycle be a loose cycle containing all vertices. Loose Hamilton cycle conjecture. For every $k\geq 3$, there is a constant $d_0=d_0(k)$ such that, for any $d\geq d_0$,

$$\mathbb H^{(k)}(n,d)\text{ contains a loose Hamilton cycle a.a.s.}$$

This conjecture extends the corresponding result for graphs, where Robinson and Wormald proved that fixed degree $d\geq 3$ suffices. For $k\geq 3$, the variance calculations required by their configuration-model approach become substantially more complicated, so the assertion remains open.