The square-free values density conjecture for polynomials at primes

Let $f\in\mathbb Z[x]$ be a polynomial with no repeated factor. Assume that for every prime $p$ there is an integer $n_p$ such that $p^2$ does not divide $f(n_p)$. Let $\rho_f(d)$ denote the number of invertible residue classes $x$ modulo $d$ satisfying $f(x)\equiv 0\pmod d$, and let $\mathcal P_{f,2}(x)$ be the set of primes $p\leq x$ for which $f(p)$ is square-free. Write $\pi(x)$ for the number of primes at most $x$.

Square-free values density conjecture. As $x\to\infty$,

$$|\mathcal P_{f,2}(x)|\sim c_{f,2}\pi(x),$$

where the positive density is

$$c_{f,2}=\prod_p\left(1-\frac{\rho_f(p^2)}{p^2-p}\right).$$

This refines the assertion that infinitely many square-free values occur at primes by predicting their natural density. The source presents it as conjectural; the supplied text gives no resolution for the full statement.