Green's inverse conjecture for the large sieve

Let $X$ be positive, and for every prime $p<\sqrt X$ let $\Omega_p\subseteq\mathbb{Z}/p\mathbb{Z}$ satisfy $|\Omega_p|=(p-1)/2$. Let $A\subseteq\{1,2,\ldots,X\}$ be obtained by sieving out the residue classes in $\Omega_p$ for $p<X$.

Green's inverse conjecture for the large sieve. Then $|A|\ll X^\epsilon$ unless $A$ is contained, apart from a set of size $X^\epsilon$, in the set of values of a quadratic polynomial

$$f(n)=an^2+bn+c.$$

The conjecture refines the inverse large-sieve problem by predicting that a sifted set can be substantially larger than the trivial bound only when it is essentially structured by the values of a quadratic polynomial.