Ore-degree Erdős Matching Conjecture for uniform hypergraphs

Let $r$, $s$, and $n$ be positive integers, and let ${\cal H}$ be an $r$-uniform hypergraph on $n$ vertices. For an $r$-edge $e$, let its Ore-degree be the sum of the degrees of the vertices in $e$, and write $\sigma_r({\cal H})$ for the minimum such sum over all edges of ${\cal H}$.

Ore-degree Erdős Matching Conjecture. If $n>rs$ and

$$\sigma_r({\cal H})>r\left(\binom{n-1}{r-1}-\binom{n-s}{r-1}\right),$$

then ${\cal H}$ contains a matching of size $s$.

The paper proves this Ore-degree analogue when $n\geq 3r^2(s-1)$ and conjectures that the same conclusion holds under the sharp-looking condition $n>rs$. The supplied context states that the conjecture remains open for $r\geq 3$.