Cross-intersection conjecture for levels of hereditary families

Let $\mathcal{H}$ be a hereditary family, let $\mathcal{H}^{(r)}$ be its $r$-th level, and write $\mu(\mathcal{H})$ for the minimum size of a base. Two subfamilies are cross-intersecting if every member of one intersects every member of the other.

Cross-intersection conjecture for hereditary families. If $\mu(\mathcal{H})\geq 2r$ and $\mathcal{A},\mathcal{B}\subseteq\mathcal{H}^{(r)}$ are cross-intersecting, then

$$|\mathcal{A}|+|\mathcal{B}|\leq |\mathcal{H}^{(r)}|.$$

The source explicitly calls this a generalization of Kamat's conjecture and does not report a resolution.