Borg's product conjecture for cross-intersecting hereditary families

Let $\mathcal{H}$ be a hereditary family, let $k\geq2$, 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 of maximum size.

Borg's product conjecture. The product

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

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

The paper notes that the weaker sum conjecture would imply this product assertion in its range, via the arithmetic mean–geometric mean inequality, but proposes the displayed statement for every hereditary family and every $k\geq2$.