The Hàn–Person–Schacht perfect matching threshold conjecture

Let $k$ and $\ell$ be integers with $1\leq\ell<k$, and let $H$ be a $k$-graph on $n$ vertices. Write $\delta_\ell(H)$ for the minimum $\ell$-degree and let $c^*_{k,\ell}$ be the fractional matching threshold defined by the condition that every $k$-graph with minimum $\ell$-degree at least $(c^*_{k,\ell}+o(1))\binom{n-\ell}{k-\ell}$ contains a perfect fractional matching. Hàn–Person–Schacht conjecture. If

$$\delta_\ell(H)\geq \left(\max\{1/2,c^*_{k,\ell}\}+o(1)\right)\binom{n-\ell}{k-\ell},$$

then $H$ contains a perfect matching. This is an asymptotic minimum-degree conjecture for perfect matchings in uniform hypergraphs. The source says that it is supported by all known results; it is known in the parameter ranges stated in the paper, but remains open in general.