Consider a union-closed family $\mathcal{F}$ of $n=\vert \mathcal{F} \vert$ sets, $n$ odd, and its family $\mathit{J}(\mathcal{F})$ of $m = \vert\mathit{J}(\mathcal{F})\vert$ basis sets (or $\cup$-irreducible sets, also called generators), and define $\mathcal{F_a} = \{A \in \mathcal{F}: a \in A \}$, $\mathit{J_a}(\mathcal{F}) = \{A \in \mathit{J}(\mathcal{F}) : a \in A \}$.

It is easy to prove by contradiction that if $\exists a$ such that $\vert\mathit{J_a}(\mathcal{F})\vert \ge m-\Bigl\lfloor\log_2{\frac{n+1}{2}}\Bigr\rfloor$, then $\vert \mathcal{F_a} \vert \ge \frac{n}{2}$.

I have tried the two examples in "The journey of the union-closed sets conjecture" by Bruhn and Schaudt on section 6.3 "Limits of averaging" and both seem to satisfy the above requirement.

I understand it might not be easy, but any idea on how to construct an example of a union-closed family with average set size less than half of the universe and such that $\forall a$ $\vert\mathit{J_a}(\mathcal{F})\vert \lt m-\Bigl\lfloor\log_2{\frac{n+1}{2}}\Bigr\rfloor$?