Polylogarithmic bilinear Bogolyubov conjecture

Let $V$ be the ambient vector space, let $P\subset V\times V$ have density $\delta>0$, and let $c(\delta)$ be the constant in Theorem 1.1, which asserts that there are subspaces $W_1,W_2\leq V$ and a family of bilinear forms on $W_1\times W_2$ satisfying the stated containment, with $\max(r_1,r_2,r_3)\leq c(\delta)$. Polylogarithmic bilinear Bogolyubov conjecture. In Theorem 1.1, one can take

$$c(\delta)=O\left(\log^{O(1)}\delta^{-1}\right).$$

The conjecture seeks a polylogarithmic bound in the density parameter for the bilinear Bogolyubov theorem, improving the triple-exponential bound stated there; the paper notes that its proof already gives such a bound for $r_1$ and $r_3$, while the symmetric role of $r_1$ and $r_2$ motivates the conjecture.