Hilton–Milner-type conjecture for nontrivial Erdős matching families

Let $n,k,s$ be positive integers, and let $\mathcal F\subset {[n]\choose k}$ be a $k$-uniform family. Write $\nu(\mathcal F)$ for its matching number and $\tau(\mathcal F)$ for its covering number. For $i\in[k]$, let

$$\mathcal A_i^{(k)}(n,s):=\Bigl\{A\in {[n]\choose k}:\bigl|A\cap[(s+1)i-1]\bigr|\ge i\Bigr\},$$

and define

$$\mathcal H^{(k)}(n,s):=\Bigl\{H\in {[n]\choose k}:H\cap[s]\ne\emptyset\Bigr\}\cup\bigl\{[s+1,s+k]\bigr\}-\Bigl\{H\in {[n]\choose k}:H\cap[s]=\{s\},\ H\cap[s+1,s+k]=\emptyset\Bigr\}.$$

Hilton–Milner-type conjecture. If $\nu(\mathcal F)=s$ and $\tau(\mathcal F)>s$, then

$$|\mathcal F|\le\max\Bigl\{\bigl\{|\mathcal A_i^{(k)}(n,s)|:i=2,\ldots,k\bigr\},|\mathcal H^{(k)}(n,s)|\Bigr\}.$$

This conjecture seeks the sharp bound for families with matching number exactly $s$ that are not covered by $s$ elements, generalizing the Hilton–Milner phenomenon; its resolution status is not specified in the source.