The discriminant-controlled polynomial congruence conjecture

Fix an integer $d\geq2$. For a monic polynomial $f(x)\in\mathbb Z[x]$, consider integers $b$ such that whenever $p^2$ divides $b$, the prime $p$ does not divide the discriminant of $f$. The polynomial congruence conjecture. For every $\epsilon>0$, there is a constant $N(d,\epsilon)$ such that there are at most $N(d,\epsilon)$ integers $n$ with

$$0\le n<b^{1-\epsilon},\qquad f(n)\equiv0\pmod b.$$

The source presents this as an open variant of the bounded-residue-set conjecture, motivated by the obstruction from repeated roots modulo prime powers.