The extremal characterization conjecture for the coefficient $f_r(s)$

Let $r$ and $s$ be the parameters in Theorem 1, and let $f_r(s)$ denote the coefficient defined there. Write $\mathcal{K}_t^r$ for the complete $r$-uniform hypergraph on $t$ vertices. For a hypergraph on $s+r$ vertices, consider removing a minimum edge set from $\mathcal{K}_{s+r}^r$ such that every $(r-1)$-set is covered at least once.

Extremal characterization conjecture. $f_r(s)$ achieves the maximum for either $\mathcal{K}_{s+r-1}^r$, or the hypergraph $\mathcal{G}$ on $s+r$ vertices obtained by this minimum-edge removal procedure. In particular, if the Steiner system $S(s+r,r,r-1)$ exists, then

$$f_r(s)=\binom{s+r}{r}-\frac{1}{r}\binom{s+r}{r-1}.$$

Here, the Steiner system $S(s+r,r,r-1)$ is a collection of $r$-element subsets of an $(s+r)$-element set such that each $(r-1)$-subset is contained in exactly one $r$-subset.

This conjecture proposes the extremal configurations determining the coefficient in the paper's general bound. The displayed value follows in the Steiner-system case; no resolution is supplied in the source.