Frankl and Kupavskii's matching–cover conjecture for uniform hypergraphs

Let $n,s,k$ be positive integers with $k\geq 2$ and $n\geq ks+k-1$. Let $H$ be a $k$-graph on vertex set $[n]$, and let $\nu(H)$ and $\tau(H)$ denote its matching number and vertex-cover number. For $2\leq i\leq k-1$, define

$$A^k_i(n,s):=\\{e\in\binom{[n]}{k}:|e\cap[(s+1)i-1]|\geq i\\}.$$

Let $S=\\{s+1,\ldots,s+k\\}$ and define

$$HM^k_{n,s}:=\\{e\in\binom{[n]}{k}:e\cap[s-1]\neq\emptyset\\}\cup\\{S\\}\cup\\{e\in\binom{[n]}{k}:s\in e,\\ e\cap S\neq\emptyset\\}.$$

For a set $U\subseteq[n]$ of size $k(s+1)-1$, let $D^k_{n,s}(U):=\\{e\in\binom{[n]}{k}:e\subseteq U\\}$, and write $D^k_{n,s}$ for this construction. Frankl and Kupavskii's conjecture. If $\nu(H)=s$ and $\tau(H)>s$, then

$$e(H)\leq\max\\{|A^k_2(n,s)|,\ldots,|A^k_{k-1}(n,s)|,|HM^k_{n,s}|,|D^k_{n,s}|\\}.$$

The listed constructions have matching number $s$ and vertex-cover number greater than $s$, so the conjecture proposes that they determine the maximum edge count among such hypergraphs. Its status is not resolved by the supplied text.