Robust Han–Zhao conjecture for loose Hamilton cycles

Let $C_{n,k,1}$ denote a loose Hamilton cycle in an $n$-vertex $k$-uniform hypergraph, where $k-1$ divides $n$. A distribution on embeddings is vertex-spread if it has the vertex-spread property defined earlier in the paper. Robust Han–Zhao conjecture. For every $k\in\mathbb N$, there exists $C=C(k)$ such that, for every sufficiently large $n$ divisible by $k-1$, every $n$-vertex $k$-uniform hypergraph $H$ satisfying

$$\delta_{k-1}(H)\geq\frac{n}{2k-2}$$

has a $(C/n)$-vertex-spread distribution on embeddings $C_{n,k,1}\hookrightarrow H$. This is proposed as a robust version of the exact codegree threshold result of Han and Zhao; the source gives no proof of the robust statement.