Approximate Duality Conjecture for finite-field bilinear bias

For a prime $p$, let $\omega$ be a $p$th root of unity with $\omega\neq1$. For subsets $A,B\subseteq\mathbb{F}_p^n$, define their bias by

$$D_\omega(A,B)=\left|\mathbb{E}_{a\sim A,\,b\sim B}[\omega^{\langle a,b\rangle}]\right|.$$

Approximate Duality Conjecture. If $D_\omega(A,B)\geq\varepsilon$ for some $\varepsilon>0$, then there exist subsets $A'\subseteq A$ and $B'\subseteq B$ such that $\langle a,b\rangle$ is constant for all $a\in A'$ and $b\in B'$, and

$$|A'||B'|\geq 2^{-O(\sqrt{n\log(1/\varepsilon)})}|A||B|.$$

The conjecture was proposed as an approach to the log-rank conjecture and is used in the paper to derive results on set systems with prescribed intersection sizes. The supplied text explicitly says that it remains open, although partial progress is known.