Kruskal-Katona type question for union-closed families of sets

Question: Let $$n,k$$ be two positive integers with $$n geq k$$. Let $$mathcal{F}$$ be a family of $$C(n,k)$$ sets, each of size $$k$$, and let $$langlemathcal{F}rangle$$ denote the union-closed family generated by $$mathcal{F}$$, i.e.: $$langlemathcal{F}rangle$$ consists of all those sets which can be expressed as a union of members of $$mathcal{F}$$.

Must it be the case that
$$begin{equation} |langlemathcal{F}rangle| geq sum_{j=k}^{n} C(n,j), end{equation}$$
with equality if and only if $$mathcal{F}$$ consists of all $$k$$-element subsets of an $$n$$-set ?

Let $$w(mathcal G)$$ denote the average size of the members of a family $$mathcal G$$.

It is easy to see that if the inequality holds (whatever about uniqueness), then it implies that, for any union-closed family $$mathcal{G}$$ and non-negative integer $$m$$,
$$|mathcal{G}| geq 2^{m}implies w(mathcal{G})ge m/2.$$
This is, in turn, a special case of a result of Reimer  that, for any union-closed family $$mathcal{G}$$ one has
$$w(mathcal{G}) geq frac{1}{2} log_{2} |mathcal{G}|.$$
Indeed I had conjectured the same result and in thinking about it was led to the above question, before I recently became aware of Reimer’s proof, which is a beautiful piece of work !

One can obviously try to generalise my question to an arbitrary number of generating $$k$$-sets, perhaps along the lines of the Kruskal-Katona theorem for shadows ?

 Reimer, David, An average set size theorem, Comb. Probab. Comput. 12, No. 1, 89-93 (2003). ZBL1013.05083.

MathOverflow Asked by Peter Hegarty on January 30, 2021

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