Frankl's binomial norm conjecture for intersecting-union families

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ denote its power set, and let an IU-family be a family $\mathcal F\subseteq 2^{[n]}$ in which any two members intersect and no two members cover $[n]$. Define its binomial norm by

$$\lVert\mathcal F\rVert_n=\sum_{F\in\mathcal F}\binom{n}{|F|}^{-1}.$$

Frankl's conjecture. For every IU-family $\mathcal F\subseteq 2^{[n]}$,

$$\lVert\mathcal F\rVert_n\leq \frac{n+1}{6}.$$

Frankl's circle-method argument gives the weaker bound $\lVert\mathcal F\rVert_n\leq n/4$; the conjectured sharp constant remains unresolved.