Packing conjecture for type-$l$ Hamilton cycles

Let $k>l\ge 2$. A random $k$-uniform hypergraph $H^k_{n,p}$ has vertex set of size $n$, and a type-$l$ Hamilton cycle is a Hamilton cycle whose consecutive edges overlap in $l$ vertices. Here $\operatorname{whp}$ means with high probability as $n\to\infty$, and $o(1)$ denotes a quantity tending to zero.

Packing conjecture. There exists a constant $C$ such that, if $(k-l)\mid n$ and

$$p \geq \frac{\log^{C} n}{n^{k-l}},$$

then, with high probability, $H^k_{n,p}$ contains

$$\frac{{n \choose k}p}{n/(k-l)}(1+o(1))$$

edge-disjoint type-$l$ Hamilton cycles.

The preceding result establishes the analogous assertion for loose Hamilton cycles, corresponding to $l=1$, up to a polylogarithmic factor. The authors explain that their proof does not extend to $l\geq 2$ because the edge exposure used to close paths becomes polynomially wasteful; obtaining this packing result for larger overlap types is posed as an interesting direction.