Extremal-family conjecture for non-star intersecting s-stable families

Let $k\geq 4$, let $\binom{[n]}{k}_{s\text{-stable}}$ be the family of $s$-stable $k$-subsets of $[n]$, and let $\mathcal{F}$ be a family in this set. An intersecting family is one whose members pairwise intersect; it is a non-star family if it is not contained in a family of all sets containing a fixed element. Let $\tau(\mathcal{F})$ denote its covering number. For $x\in[n]$ and $Y\in\binom{[n]}{k}_{s\text{-stable}}$ with $x\notin Y$, define

$$\mathcal{F}_{s,k}(x,Y)=\{Y\}\cup\{F\in\binom{[n]}{k}_{s\text{-stable}}\mid x\in F,\ F\cap Y\neq\emptyset\},$$

and let $\mathcal{G}_{s,k}$ be the collection of all such families.

Extremal-family conjecture. If $\mathcal{F}\subseteq\binom{[n]}{k}_{s\text{-stable}}$ is a non-star intersecting family with $\tau(\mathcal{F})\geq 2$, then, for $n$ large enough—suggested in the source as $n\geq k(2s-1)+1$—

$$|\mathcal{F}|\leq\max\{|F|\mid F\in\mathcal{G}_{s,k}\}.$$

This is an extremal set-theoretic conjecture intended to identify the largest non-star intersecting families of $s$-stable sets. The source gives only a suggested threshold and does not provide a resolution.