The empty-family extremal conjecture for hereditary levels

Let $\mathcal{H}$ be a hereditary family, let $\mathcal{H}^{(r)}$ and $\mathcal{H}^{(s)}$ be its levels, and suppose $\mathcal{A}\subseteq\mathcal{H}^{(r)}$ and $\mathcal{B}\subseteq\mathcal{H}^{(s)}$ are cross-$t$-intersecting, meaning every $A\in\mathcal{A}$ and $B\in\mathcal{B}$ satisfy $|A\cap B|\geq t$. Empty-family extremal conjecture. If $1\leq t\leq r\leq s$ and

$$\mu(\mathcal{H})\geq r+s-t+1,$$

then

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

Equivalently, when empty families are allowed, the maximum is attained by $\mathcal{A}=\emptyset$ and $\mathcal{B}=\mathcal{H}^{(s)}$. This is presented as a strong generalization of Kamat's conjecture and remains open in the source.