A conjecture related to Frankl's conjecture

Question

Let $\mathcal{F}\subseteq2^{[n]},\emptyset\in\mathcal{F}$ be an union-closed family of sets. For $S\in\mathcal{F}$, let $w(S)$ be the number of subsets of $S$ in $\mathcal{F}$. Does there always exist real numbers $a_1,a_2,...,a_n\geq1$ such that $\prod_{i=1}^na_i=|\mathcal{F}|$ and for every $S\in\mathcal{F}$, $\prod_{i\in S}a_i\geq w(S)$?

Motivation: Let $w_i$ be the number of sets of $\mathcal{F}$ that contain $i$. If the above conjecture true, we have $\sum_{i=1}^n\ln a_iw_i\geq\sum_{S\in\mathcal{F}}\ln w(S)$, so we just have to prove $\sum_{S\in\mathcal{F}}\ln w(S)\geq\sum_{i=1}^n\ln a_i\frac{|\mathcal{F}|}{2}=\frac{|\mathcal{F}|\ln|\mathcal{F}|}{2}$ that would imply Frankl's conjecture.

Answer 1

so we just have to prove $\sum_{S\in\mathcal{F}}\ln w(S)\geq \dots\frac{|\mathcal{F}|\ln|\mathcal{F}|}{2}$

This inequality does not depend on the choice of $a_i$'s, and, unfortunately, it can fail. For instance, let $n=3$ and $\mathcal F=\{\emptyset,\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$). Then $|\mathcal F|=5$, $\sum_{S\in\mathcal{F}}\ln w(S)=\ln 1+3\ln 2+\ln 5$ and $\frac{|\mathcal{F}|\ln|\mathcal{F}|}{2}=\frac{5\ln 5}{2}$. But $$\ln 1+3\ln 2+\ln 5-\frac{5\ln 5}{2}=3\cdot\left(\ln 2-\frac{\ln 5}{2}\right)<0.$$