Green--Harper's inverse quadratic large sieve conjecture

Let $\mathcal A$ be a set of positive integers. For each prime $p$, let $\mathcal A_p$ denote the set of residue classes occupied by $\mathcal A$ modulo $p$. A rational quadratic is a quadratic polynomial with rational coefficients. Green--Harper's conjecture. If

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

for all sufficiently large primes $p$, then either there is a rational quadratic $\psi$ such that all but finitely many elements of $\mathcal A$ lie in $\psi(\mathbb Q)$, or, for every integer $k$, there are arbitrarily large values of $X$ such that

$$|\mathcal A\cap [X]|<\frac{X^{1/2}}{(\log X)^k}.$$

Green and Harper's three-alternative theorem established a partial result: the remaining alternative was that the set had very small intersection with every fixed rational quadratic. Their conjecture asserts that this third alternative is unnecessary; the stated dichotomy remains open.