Inverse large sieve stability conjecture for finite sets

Let $X_0\in\mathbb{N}$, let $\rho>0$, and let $X\in\mathbb{N}$ be sufficiently large in terms of $X_0$ and $\rho$. Suppose that $A,B\subset[X]$ satisfy

$$|A\pmod p|+|B\pmod p|\leqslant p+1$$

for all $p\in[X_0,X^{1/4}]$. Inverse large sieve stability conjecture. There exists a constant $c=c(\rho)>0$ such that either

$$|A\cap[X^{1/2}]|\leqslant X^{1/4-c}\quad\text{or}\quad |B\cap[X^{1/2}]|\leqslant X^{1/4-c},$$

or there are rational quadratics $\psi_A,\psi_B$ of height at most $X^\rho$ such that

$$|A\setminus\psi_A(\mathbb{Q})|\leqslant X^{1/2-c}\quad\text{and}\quad |B\setminus\psi_B(\mathbb{Q})|\leqslant X^{1/2-c}.$$

This conjecture asserts that near-extremal examples for the large sieve either have a substantially smaller contribution below $X^{1/2}$ or possess quadratic structure of controlled height; the preceding stability theorem proves a parameterized finite analogue, while the conjectural uniform statement remains open.