Zeon formulation of Frankl's union-closed sets conjecture

Let $H$ be a hypergraph with vertex set $V=\{v_1,\ldots,v_n\}$ and hyperedge set satisfying condition $\mathfrak{F}$, meaning that the union of any two hyperedges is again a hyperedge. Let $\Gamma_H$ be the zeon incidence representation of $H$, and let $\langle\langle\cdot\rangle\rangle$ denote the quantity used in the source to count the corresponding hyperedges.

Zeon formulation of Frankl's conjecture. There exists $i\in[n]$ such that

$$\langle\langle\zeta_i\Gamma_H\rangle\rangle\leq \frac{1}{2}\langle\langle\Gamma_H\rangle\rangle.$$

This is presented as the zeon-incidence formulation equivalent to Frankl's union-closed sets conjecture. It remains open.