The squarefree-values conjecture for polynomials

Let $f(x)\in\mathbb{Z}[x]$ have no multiple roots in $\mathbb{C}$. Let $B$ be the largest integer dividing $f(n)$ for every integer $n$, and let $B'$ be the smallest divisor of $B$ such that $B/B'$ is squarefree.

Squarefree-values conjecture. Then $f(n)/B'$ is squarefree for infinitely many values of $n$.

This is a folklore conjecture on squarefree values of polynomial sequences. The source states that it is known when $\deg f\leq 3$ by Hooley and that it follows from the abc conjecture; it is used to improve the paper's class-number result.