Random hypergraph transference conjecture for perfect matchings

Fix positive integers $d<k$, $nobreak$fix $\gamma>0$, and let $G\sim H^k(n,p)$, where $0<p<1$ may depend on $n$ and $k\mid n$. Here $H^k(n,p)$ is the random $k$-uniform hypergraph on $n$ vertices in which each $k$-set is included independently with probability $p$, and a spanning subgraph $G'\subseteq G$ has minimum $d$-degree $\delta_d(G')$ equal to the minimum number of edges of $G'$ containing any fixed $d$-set. The quantity $\mu_d(k)$ is the limiting normalized Dirac threshold defined by

$$\mu_d(k)=\lim_{n\to\infty,\ k\mid n}\frac{m_d(k,n)}{\binom{n-d}{k-d}}.$$

Random transference conjecture. Asymptotically almost surely, every spanning subgraph $G'\subseteq G$ satisfying

$$\delta_d(G')\geq (\mu_d(k)+\gamma)\binom{n-d}{k-d}p$$

has a perfect matching. This conjecture proposes a sparse random analogue of the deterministic Dirac-type theorem; it is intended to transfer minimum-degree results to random hypergraphs, and remains open in the stated generality.