Borg's strong form for extremal cross-intersecting levels

Let $\mathcal{H}$ be a hereditary family, with $\mathcal{H}^{(r)}$ and $\mathcal{H}^{(s)}$ its $r$th and $s$th levels. For cross-intersecting subfamilies, let $M(\mathcal{H}^{(r)},\mathcal{H}^{(s)},1)$ denote the pairs maximizing the sum of their sizes. A family is trivial $1$-intersecting if all its members contain a common element. Borg's weak form. If $1 \leq r \leq s$ and

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

then for some $(\mathcal{A},\mathcal{B}) \in M(\mathcal{H}^{(r)},\mathcal{H}^{(s)},1)$, $\mathcal{A}$ is a trivial $1$-intersecting family. The source reports that this weak form is disproved by Borg's result, which gives counterexamples for $2 \leq r \leq s$ and suitable hereditary families.