Mészáros–Rónyai conjecture on deleting a member of an s-extremal family

For a family $\mathcal{F}\subseteq 2^{[n]}$, let $\operatorname{tr}(\mathcal{F})$ be its trace family. The family is s-extremal when

$$|\operatorname{tr}(\mathcal{F})|=|\mathcal{F}|.$$

Mészáros–Rónyai conjecture. For every nonempty s-extremal family $\mathcal{F}\subseteq 2^{[n]}$, there exists $F_0\in\mathcal{F}$ such that $\mathcal{F}\setminus\{F_0\}$ is still s-extremal.

The conjecture is known for hereditary families and for all families with VC dimension at most $2$; it remains open in general.