Weighted Frankl conjecture for union-closed families

Let $[n]=\{1,\ldots,n\}$, and let $\mathcal{F}\subseteq 2^{[n]}$ be a non-empty union-closed family, meaning that $A\cup B\in\mathcal{F}$ for all $A,B\in\mathcal{F}$. Let $x_1,x_2,\ldots,x_n\geq1$ be real numbers. Weighted Frankl conjecture. There exists $k\in[n]$ such that

$$\sum_{S\in\mathcal{F},\ k\in S}\prod_{m\in S,\ m\neq k}x_m\geq\sum_{S\in\mathcal{F},\ k\notin S}\prod_{m\in S}x_m.$$

This weighted assertion is motivated by blowing up each element into a family of non-empty subsets. The source presents it as a new conjecture derived from Frankl's conjecture, and gives no resolution; in particular, its general validity remains open.