Kühn–Osthus–Townsend conjecture on minimum $d$-degree thresholds for matchings

Let $H$ be a $k$-graph, let $d$ be an integer with $1\leq d\leq k-1$, and let $m_d^s(k,n)$ denote the minimum integer $m$ such that every $n$-vertex $k$-graph $H$ with minimum $d$-degree at least $m$ contains a matching of size $s$. For any $\varepsilon>0$, with $0\leq s\leq(1-\varepsilon)n/k$, Kühn–Osthus–Townsend conjecture.

$$m_d^s(k,n)=\left(1-\left(1-\frac{s}{n}\right)^{k-d}+o(1)\right)\binom{n-d}{k-d}.$$

This conjecture predicts the asymptotic minimum codegree-type threshold forcing a matching of size $s$. The source introduces it as a conjecture of Kühn, Osthus, and Townsend; its resolution status is not specified in the supplied text.