The covering-number weakening for resilient hypergraph extremal size

Let $\mathcal{F}$ be a $k$-uniform hypergraph, let $\nu(\mathcal{F})$ be its matching number, and let $\tau(\mathcal{F})$ be its covering number, the minimum size of a vertex set meeting every edge. Resilient hypergraph covering conjecture. If $\nu(\mathcal{F})=s$ and $\tau(\mathcal{F})=sk$, then, for $s>s_0(k)$,

$$|\mathcal{F}|\leq\binom{sk+k-1}{k}.$$

This is presented as a weaker version of the preceding conjecture and remains open in the stated range.