Cross-intersecting set-system conjecture

Let $\mathcal A,\mathcal B\subseteq 2^{[d]}$. For $L\subseteq\{0,\ldots,d\}$, they are $L$-cross-intersecting if $|A\cap B|\in L$ for every $A\in\mathcal A$ and $B\in\mathcal B$.

Cross-intersecting set-system conjecture. For every fixed $k>0$, if $|L|=k$ and $\mathcal A,\mathcal B$ are $L$-cross-intersecting, then there exist $\mathcal R\subseteq\mathcal A$, $\mathcal S\subseteq\mathcal B$, and $t\in L$ such that $\mathcal R$ and $\mathcal S$ are $\{t\}$-cross-intersecting and

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

Equivalently, the matrix with entries $|A\cap B|$ contains a monochromatic rectangle of density at least $2^{-\operatorname{polylog}(d)}$. The source says this conjecture would be implied by the parallel $k$-partition conjecture and is therefore equivalent to the log-rank conjecture, but it remains open.