The asymptotic half-frequency conjecture for the kth-most frequent element

Let $\mathcal F$ be a union-closed set family, let $f_k(\mathcal F)$ denote its $k$th-highest element frequency, and fix $k\in\mathbb N$. Asymptotic half-frequency conjecture. As $|\mathcal F|\to\infty$,

$$f_k(\mathcal F)=\frac12-o(1).$$

The paper establishes the weaker asymptotic lower bound $f_k(\mathcal F)\geq(3-\sqrt5)/2-o(1)$; whether the $k$th-most frequent element asymptotically appears in half the sets remains open.