Average-rarity conjecture for rooted-set representations

Let $V$ be a finite ground set, and let $\mathcal C$ be an intersection-closed family on $V$ containing $\varnothing$. Suppose that $\mathcal C$ admits a rooted-set representation in which every element of $V$ is the root of at most one rooted set. Average-rarity conjecture. Then $\mathcal C$ is average-rare.

The paper proves average-rarity for order-ideal families of functional preorders, a special case in which the rooted-set stems are singletons. The conjecture extends this result to rooted sets with larger stems; existence of a rare element is known in the relevant general setting, but the stronger average-rarity assertion remains open.