Regular hypergraph overlapping Hamilton cycle conjecture

Let $\mathbb H^{(k)}(n,d)$ denote the random $d$-regular $k$-uniform hypergraph on $n$ vertices. For integers $k>\ell\geq 2$, an $\ell$-overlapping cycle is a $k$-uniform hypergraph whose edges arise from a cyclic vertex ordering, with each edge consisting of $k$ consecutive vertices and each pair of consecutive edges sharing exactly $\ell$ vertices. Overlapping Hamilton cycle conjecture. For every $k>\ell\geq 2$, if $d\gg n^{\ell-1}$, then

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

Thresholds for $\ell$-overlapping Hamilton cycles in random hypergraphs with independent edges are known, but the analogous result for random regular hypergraphs with arbitrary $\ell\geq 2$ remains open.