Diversity bound for intersecting-free hypergraph families

Let $\mathcal F\subset {[n]\choose k}$ satisfy $\nu(\mathcal F)\leq s$ and $n\geq k(s+1)$. Define the $s$-diversity by

$$\gamma_s(\mathcal F)=\min_T|\mathcal F(\bar T)|,$$

where $\mathcal F(\bar T)=\\{F\in\mathcal F:F\cap T=\emptyset\\}$. Diversity bound conjecture. One has

$$\gamma_s(\mathcal F)\leq\max\left\{\sum_{l=2}^{s+1}{{s+1}\choose l}{{n-2s-1}\choose{k-l}},{{(k-1)(s+1)}\choose k}\right\}.$$

The conjecture asks for a sharp upper bound on the diversity of a family with matching number at most $s$. The source presents it as a question for further progress; no resolution status is supplied.