The extremal cover-size conjecture for unstable hypergraph Kneser colorings

Let $k\geq 5$, and let $r$ and $\ell$ be positive integers satisfying

$$\ell<r<2\ell.$$

For a cover $C$ with elements $t_i$, let $H_{C,r}(n)$ denote the associated hypergraph, and let $\kappa(H,k,\ell)$ be its number of $(k,\ell)$-Kneser colorings. Extremal cover-size conjecture. If

$$\kappa(H,k,\ell)=\operatorname{KC}(n,r,k,\ell),$$

then

$$|C|=c(k)=\lceil k/3\rceil$$

and

$$|t_i\cap t_j|=2\ell-r-1$$

for every distinct $t_i,t_j\in C$. The conjecture concerns the unstable range, where the previously defined hypergraph $H_{n,r,k,\ell}$ is asymptotically optimal but is not extremal for sufficiently large $n$.