Local avoidance conjecture for polynomial images of integers and primes

Let $h\in\mathbb{Z}[x]$ be a nonzero polynomial, and let $X=h(\mathbb{Z})$ or $X=h(\mathcal{P})$, where $\mathcal{P}$ denotes the set of primes. For each $m\in\mathbb{N}$, let $X_m$ be the set of congruence classes modulo $m$ that intersect $X$, and define

$$d_X(m)=\max\{|A|: A\subseteq\mathbb{Z}/m\mathbb{Z},\ (A-A)\cap X_m=\emptyset\}.$$

Here $\mu(X)$ denotes the maximal density of an $X$-set, namely a set whose pairwise differences avoid $X$. Local avoidance conjecture. For every such $h$ and $X$, we have

$$\mu(X)=\sup_{m\in\mathbb{N}}\frac{d_X(m)}{m}.$$

The conjecture asserts that for polynomial images of the integers and primes, the maximal global density of a set avoiding $X$-differences is determined by the strongest finite-modulus local avoidance obstruction. It extends the principle that local obstructions are the only obstructions in intersectivity questions; the supplied context does not state whether it is known or open.