Frankl and Wang's sturdiness conjecture for IU-families

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ denote the power set of $[n]$, and let an IU-family be a family $\mathcal{G}\subseteq 2^{[n]}$ such that

$$F\cap F'\neq\emptyset\quad\text{for all }F,F'\in\mathcal{G},$$

and

$$F\cup F'\neq[n]\quad\text{for all }F,F'\in\mathcal{G}.$$

Write $\beta(\mathcal{G})$ for the sturdiness of $\mathcal{G}$. Frankl and Wang's conjecture. If $\mathcal{G}\subseteq 2^{[n]}$ is an IU-family, then

$$\beta(\mathcal{G})\leq 2^{n-4}.$$

The conjecture proposes a sharp upper bound on the local robustness, or sturdiness, of IU-families; the preceding IU-Theorem gives the corresponding maximum-size bound $|\mathcal{G}|\leq 2^{n-2}$. Its resolution is not indicated in the supplied text.