Inverse conjecture for polynomially concentrated bilinear forms

Let $A$ be an $n\times n$ matrix of nonzero entries, and let $x,y$ be chosen independently and uniformly from $\{-1,1\}^n$. For a matrix decomposition $A=A_1+A_2+A_3$, say that $A_1$ is a low-rank component, $A_2$ is a generalized-arithmetic-progression component, and $A_3$ is a sparse component according to the bounds below. Bilinear inverse conjecture. For any $a>0$, there are constants $a_1,a_2,a_3$ and $N_0$ such that, for all $n>N_0$, if

$$\sup_c \mathbb{P}(x^TAy=c)>n^{-a},$$

then $A=A_1+A_2+A_3$, where $A_1$ has rank at most $a_1$, the entries of $A_2$ are drawn from a generalized arithmetic progression of rank at most $a_2$ and volume at most $a_3$, and $A_3$ contains at most $n^2/\log n$ nonzero entries. The conjecture proposes that algebraic low rank and arithmetic structure, together with combinations of these and a sparse error, are essentially the only sources of polynomial concentration for bilinear forms.