Borg's two-configuration sum conjecture for cross-intersecting hereditary families

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 $\mathcal{H}$. Let $\mathcal{S}$ be a largest star of $\mathcal{H}$.

Borg's stronger sum conjecture. (i) If

$$k\leq\frac{|\mathcal{H}|}{|\mathcal{S}|},$$

then $\sum_{i=1}^k|\mathcal{A}_i|$ is maximum when $\mathcal{A}_1=\mathcal{H}$ and $\mathcal{A}_2=\cdots=\mathcal{A}_k=\emptyset$. (ii) If

$$k\geq\frac{|\mathcal{H}|}{|\mathcal{S}|},$$

then the sum is maximum when $\mathcal{A}_1=\cdots=\mathcal{A}_k=\mathcal{S}$.

This strengthens the preceding sum conjecture by proposing the optimal configuration on both sides of the threshold $|\mathcal{H}|/|\mathcal{S}|$. The paper explains that the stronger statement implies the $k\geq n+1$ version.