Ahlswede–Khachatrian conjecture on hypergraphs without perfect fractional matchings

Let $H(n,k)=([n],E)$ be a $k$-uniform hypergraph, where $[n]=\{1,\ldots,n\}$, and let a perfect fractional matching be a collection of nonnegative real numbers $(\alpha_e)_{e\in E}$ satisfying

$$\sum_{e\in E}\alpha_e=1,\qquad \sum_{e\in E}\alpha_e e=\left(\frac{k}{n},\ldots,\frac{k}{n}\right).$$

Let $\mathcal{H}_f$ be the set of hypergraphs without perfect fractional matchings, and define $p(n,k)=\max_{H\in\mathcal{H}_f}|E|$. Ahlswede–Khachatrian conjecture.

$$p(n,k)=\max_{k\ge s\ge 1}\sum_{i=0}^{k-s}\binom{n_s}{i+s}\binom{n-n_s}{k-s-i},$$

where $n_s=\lceil ns/k\rceil-1$. The conjecture gives an extremal formula for the largest number of edges in a $k$-uniform hypergraph lacking a perfect fractional matching. The source states that it is proved up to finitely many cases that can be checked computationally; the remaining cases are therefore open in the supplied text.