Symmetric inverse large sieve conjecture for sets of integers

Let $\mathscr{A}$ be a set of positive integers satisfying

$$|\mathscr{A}\pmod p|\leqslant\frac12(p+1)$$

for all sufficiently large primes $p$. Symmetric inverse large sieve conjecture. Either there is a rational quadratic $\psi$ such that all but finitely many elements of $\mathscr{A}$ lie in $\psi(\mathbb{Q})$, or, for every integer $k$, there are arbitrarily large $X$ such that

$$|\mathscr{A}[X]|<\frac{X^{1/2}}{\log^k X}.$$

In particular, $\liminf_{X\to\infty}X^{-1/2}|\mathscr{A}[X]|=0$. This is presented as a cleaner inverse large sieve formulation, obtained by conjecturally removing the third alternative from the preceding theorem; it remains open.