Conjecture beyond Hamilton connectivity for hypergraph cycle embeddings

For a positive real number $\alpha$ and $k\in\mathbb N$, let $\delta^{\mathrm{CON}}_{k,\ell,d}$ be the threshold appearing in the main vertex-spread theorem, let $\delta_{\mathcal{C}_{k,\ell},d}$ be the Hamilton-cycle minimum $d$-degree threshold, and let $C_{n,k,\ell}$ denote the corresponding Hamilton $\ell$-cycle. A distribution on embeddings is vertex-spread if it has the spread property defined earlier in the paper. The beyond-Hamilton-connectivity conjecture. For every $\alpha>0$ and $k\in\mathbb N$, there exists $C=C(\alpha,k)$ such that, for every $\ell,d\in[k-1]$ and every sufficiently large $n$ divisible by $k-\ell$, every $n$-vertex $k$-uniform hypergraph $H$ satisfying

$$\delta_d(H)\geq\left(\delta_{\mathcal{C}_{k,\ell},d}+\alpha\right)\binom{n-d}{k-d}$$

has a $(C/n)$-vertex-spread distribution on embeddings $C_{n,k,\ell}\hookrightarrow H$. If true, this would identify the Hamilton-cycle threshold with the robust and vertex-spread thresholds. The source notes that the claim holds for $\ell=0$, while the cases with $\ell>0$ appear difficult; no general proof is known.