The smaller matching threshold conjecture

Let $m_d^s(k,n)$ be the smallest integer $m$ such that every $k$-graph $H$ on $n$ vertices with minimum $d$-degree at least $m$ contains a matching of size $s$. Let $H_s^0(n,k)$ be the $k$-graph whose vertex set is partitioned into $A\cup B$, with $|A|=s-1$, and whose edges are all $k$-sets meeting $A$.

Smaller matching threshold conjecture. Given $1\leq d\leq k-2$, there exist $n_0$ and $C$ such that

$$m_d^s(k,n)=\binom{n-d}{k-d}-\binom{n-s-d+1}{k-d}+1$$

for all $n\geq n_0$ and all $s\leq n/k-C$.

The displayed value is exactly the lower bound supplied by the space barrier $H_s^0(n,k)$. The claim was known in some cases and an asymptotic version had been proved in a substantial range, but the stated general result remained open in the survey.