Squarefree-value residue class conjecture

Let $f(x)\in\mathbf Z[x]$ be a separable nonconstant polynomial. For a nonzero rational number $r$, let $S(r)$ be the unique squarefree integer such that $r/S(r)$ is a square in $\mathbf Q$, and define

$$\mathcal S(f)=\{S(f(r)):r\in\mathbf Q\text{ and }f(r)\ne0\}.$$

Squarefree-value residue class conjecture. For all but finitely many primes $p$, the set $\mathcal S(f)$ contains infinitely many elements from every nonzero residue class modulo $p$. This conjecture predicts that, apart from finitely many primes, squarefree parts of rational polynomial values are equidistributed in the weak sense that every nonzero residue class occurs infinitely often. The paper provides evidence for the conjecture in degrees at most four and studies the absence of local obstructions in higher degree.