Large matching minimum degree conjecture for uniform hypergraphs

Let $G$ be a $k$-uniform hypergraph on $n$ vertices. For integers $n$, $d$, $k$, and $s$ with $1\leq d\leq k-1$ and $0\leq s\leq (1-\varepsilon)n/k$, let $m_d^s(k,n)$ denote the minimum integer $m$ such that every $k$-uniform hypergraph on $n$ vertices with minimum $d$-degree at least $m$ has a matching of size $s$. Large matching degree conjecture. For every $\varepsilon>0$,

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

This extends the perfect-matching threshold problem to smaller matchings. The paper presents it as a proposed generalization; the supplied text gives no resolution of the conjecture, although it records exact results in some small-uniformity cases.