Quadratic inverse conjecture for the large sieve

Let $A\subset [1,N]$ be a set of integers. For each prime $p$, write $A_p$ for the set of residue classes modulo $p$ represented by elements of $A$. The notation $X\ll Y$ means that $|X|\leq c|Y|$ for some constant $c$, or equivalently $X=O(Y)$. Quadratic inverse conjecture for the large sieve. If

$$|A_p|\leq (p+1)/2$$

for each prime $p$ and $|A|\gg N^{1/2}$, then there is a subset $A'\subseteq A$ with

$$|A'|\geq \frac{9}{10}|A|$$

such that $A'\subseteq q(\mathbb Z)$ for some quadratic polynomial $q(x)\in\mathbb Q[x]$. The larger sieve shows that the square-root scale is the natural threshold, while the conjecture asserts that dense sets attaining this scale under the residue-class restriction must essentially come from quadratic sequences.