Almost cross-disjoint set-system conjecture

Let $\mathcal A,\mathcal B\subseteq 2^{[d]}$. They are $\epsilon$-almost cross-disjoint if

$$\Pr_{A\sim\mathcal A,\,B\sim\mathcal B}[A\cap B\neq\emptyset]\leq\epsilon.$$

They are exactly cross-disjoint when no such intersection occurs.

Almost cross-disjointness conjecture. For every fixed $\epsilon>0$, if $\mathcal A$ and $\mathcal B$ are $\epsilon$-almost cross-disjoint, then there exist $\mathcal R\subseteq\mathcal A$ and $\mathcal S\subseteq\mathcal B$ that are exactly cross-disjoint and satisfy

$$|\mathcal R||\mathcal S|\geq |\mathcal A||\mathcal B|\cdot 2^{-O(\sqrt d)}.$$

Equivalently, the intersection matrix with entries $|A\cap B|$ contains a zero-monochromatic rectangle of density at least $2^{-O(\sqrt d)}$. The source says this conjecture would follow from the incidence-density conjecture; it is not resolved in the supplied text.