Lovett–Singer–Sudan conjecture on cross-intersecting set systems

Let $\mathcal{S}=\{S_1,\ldots,S_m\}$ and $\mathcal{T}=\{T_1,\ldots,T_n\}$ be families in $2^{[d]}$, and suppose that $(\mathcal{S},\mathcal{T})$ is $\{a,b\}$-cross-intersecting, meaning that $|S\cap T|\in\{a,b\}$ for every $S\in\mathcal{S}$ and $T\in\mathcal{T}$, where $a,b\in\{0,\ldots,d\}$. Lovett–Singer–Sudan conjecture. There exist subfamilies $\mathcal{A}\subseteq\mathcal{S}$ and $\mathcal{B}\subseteq\mathcal{T}$ such that $(\mathcal{A},\mathcal{B})$ is either $\{a\}$- or $\{b\}$-cross-intersecting, and

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

This conjecture was independently proposed by Lovett and by Singer–Sudan, and the source says it is implied by the log-rank conjecture. The paper proves an equivalence between the two conjectures, so its resolution remains open.