The following question is motivated by the recent breakthrough result by Justin Gilmer on the union-closed sets (aka Frankl) conjecture.

Let $\mathcal{F}\subseteq\mathcal{P}(\mathbb{N})$ be a finite, union-closed family, i.e. $A,B\in\mathcal{F}\Rightarrow A\cup B\in\mathcal{F}$. Let $A_{\mathcal{F}}$ and $B_{\mathcal{F}}$ be two independent random variables, uniformly distributed over the elements of $\mathcal{F}$. Denoting by $H$ the entropy, we clearly have $$H(A_{\mathcal{F}}\cup B_{\mathcal{F}})\leq H(A_{\mathcal{F}})=\ln|\mathcal{F}|,$$ since the entropy is maximized by the uniform distribution. I am wondering whether a sharper bound of the form $H(A_{\mathcal{F}}\cup B_{\mathcal{F}})\leq \lambda H(A_{\mathcal{F}})$, for some $\lambda<1$, still continue to hold. The intuition is that the distribution $A_{\mathcal{F}}\cup B_{\mathcal{F}}$ should "deviate" enough from the uniform bound in order to get a non-trivial upper bound on its entropy. Denoting by $\mathrm{UC}$ the collection of all finite, union-closed families $\mathcal{F}\subseteq\mathcal{P}(\mathbb{N})$ (with $|\mathcal{F}|>1$), we then consider the following quantity:

$$\lambda:=\sup_{\mathcal{F}\in UC}\frac{H(A_{\mathcal{F}}\cup B_{\mathcal{F}})}{H(A_{\mathcal{F}})}=\sup_{\mathcal{F}\in UC}\frac{H(A_{\mathcal{F}}\cup B_{\mathcal{F}})}{\ln|\mathcal{F}|}.$$

We have $\lambda\leq 1$, and by considering the union-closed families $\mathcal{P}[n]\setminus\{\emptyset\}$, we get the lower bound $\lambda\geq 0.82$.

**Is $\lambda=1$ or $\lambda<1$? In the latter case, it is possible to provide an explicit, non-trivial upper bound?**