The finitary inverse large sieve conjecture

Let $X\subset [N]$ be a subset and let $\epsilon>0$ be real. Assume that, for every prime $p\geq\epsilon^{-1}$, the reduction $X\pmod p$ occupies at most $0.99p$ residue classes. Finitary inverse large sieve conjecture. Either $|X|\ll_{\epsilon}N^{\epsilon}$, or there exists a polynomial $f(x)\in\mathbb{Q}[x]$ of degree $2\leq d=O_{\epsilon}(1)$ and height $N^{O_{\epsilon}(1)}$ such that $|X\cap f(\mathbb{Q})|\gg_{\epsilon}|X|$. This finitary formulation is proposed after the infinitary version is shown to be false; the required polynomial height bounds are noted as an unresolved technical issue, and the conjecture is used as the hypothesis for improved larger-sieve estimates.