Robust exact-threshold conjecture for Hamilton cycle embeddings

For $k\in\mathbb N$, $d\in[k-1]$, $\ell\in\{0,\dots,k-1\}$, and $n$ divisible by $k-\ell$, let $h_{k,\ell,d}(n)$ be the least integer $D\geq0$ such that every $n$-vertex $k$-uniform hypergraph $H$ with $\delta_d(H)\geq D$ contains a Hamilton $\ell$-cycle. Let $C_{n,k,\ell}$ denote that cycle, and call a distribution on its embeddings vertex-spread when it satisfies the paper's vertex-spread condition. Robust exact-threshold conjecture. For every $k\in\mathbb N$, there exists $C=C(k)$ such that, for every $d\in[k-1]$, $\ell\in\{0,\dots,k-1\}$, and admissible $n$, every $n$-vertex $k$-uniform hypergraph $H$ with

$$\delta_d(H)\geq h_{k,\ell,d}(n)$$

has a $(C/n)$-vertex-spread distribution on embeddings $C_{n,k,\ell}\hookrightarrow H$. This is a robust strengthening of the exact Hamilton-cycle threshold statement. The source presents it as open, although it records some special cases where the underlying exact threshold is known.