A *Frankl family* is a nonempty finite family $\mathcal F$ of nonempty finite sets such that $A,B\in\mathcal F\implies A\cup B\in\mathcal F.$ Define $d_\mathcal F(x)=|\{A\in\mathcal F:x\in A\}|$ and $\Delta(\mathcal F)=\max_xd_\mathcal F(x).$ Frankl's union-closed sets conjecture says that $\Delta(\mathcal F)\gt\frac12|\mathcal F|,$ or in other words $|\mathcal F|\le2\Delta(F)-1,$ for any Frankl family $\mathcal F.$ Has the following stronger conjecture been considered, and is a counterexample known?

For any Frankl family $\mathcal F$ there is an element $x$ such that $\Delta(\{A\in\mathcal F:x\notin A\})\le\frac12\Delta(\mathcal F),$

or better yet,

for any Frankl family $\mathcal F$ and any element $x,$ if $d_\mathcal F(x)=\Delta(\mathcal F)$ then $\Delta(\{A\in\mathcal F:x\notin A\})\le\frac12\Delta(\mathcal F)$?