  The union-closed sets conjecture states that if a family of sets $\mathcal{A}
\neq \{\emptyset\}$ is union-closed, then there is an element which belongs to
at least half the sets in $\mathcal{A}$. In 2001, D. Reimer showed that the
average set size of a union-closed family, $\mathcal{A}$, is at least
$\frac{1}{2} \log_2 |\mathcal{A}|$. In order to do so, he showed that all
union-closed families satisfy a particular condition, which in turn implies the
preceding bound. Here, answering a question raised in the context of T. Gowers'
polymath project on the union-closed sets conjecture, we show that Reimer's
condition alone is not enough to imply that there is an element in at least
half the sets.
