Borg's sum conjecture for cross-intersecting hereditary subfamilies

Let $[n]=\{1,\ldots,n\}$, let $\mathcal{H}\neq\{\emptyset\}$ be a hereditary subfamily of $2^{[n]}$, and let $\mathcal{A}_1,\ldots,\mathcal{A}_k$ be cross-intersecting subfamilies of $\mathcal{H}$. A largest star $\mathcal{S}$ of $\mathcal{H}$ is a star having maximum size among the stars of $\mathcal{H}$.

Borg's sum conjecture. If $k\geq n+1$, then

$$\sum_{i=1}^k|\mathcal{A}_i|$$

is maximum when $\mathcal{A}_1=\cdots=\mathcal{A}_k=\mathcal{S}$ for some largest star $\mathcal{S}$ of $\mathcal{H}$.

The threshold $k\geq n+1$ is necessary in the examples given in the paper, and the conjecture is proved there for important classes of hereditary families, including families compressed with respect to an element.