The extremal-function formulation of Frankl's conjecture

For $a,n\in\mathbb{N}^+$, let $\mathcal{S}\subseteq 2^{[n]}$ be a union-closed family such that every element of $[n]$ belongs to at most $a$ sets of $\mathcal{S}$. Let $f(n,a)$ denote the maximum possible cardinality of such a family.

Extremal formulation of Frankl's conjecture. For every $a,n\in\mathbb{N}^+$,

$$|\mathcal{S}|\leq 2a.$$

Equivalently, $f(n,a)\leq 2a$ for all $a,n\in\mathbb{N}^+$. This formulation is stated as equivalent to Frankl's conjecture and is the starting point for the paper's linear-relaxation bounds, but it remains open in general.