Robust Katona–Kierstead conjecture for 3-uniform tight cycles

Let $H$ be an $n$-vertex $3$-uniform hypergraph, and let $C_{n,3,2}$ denote its tight Hamilton cycle. A distribution on embeddings is vertex-spread if it has the vertex-spread property defined earlier in the paper. Robust Katona–Kierstead conjecture. There exists an absolute constant $C$ such that, whenever

$$\delta_2(H)\geq\left\lfloor\frac n2\right\rfloor,$$

there is a $(C/n)$-vertex-spread distribution on embeddings $C_{n,3,2}\hookrightarrow H$. The source places this proposed special case in an ignored conditional section and gives no resolution, so it remains open as stated.