Squarefree values of polynomials at primes

Let $f(t)\in\mathbb{Z}[t]$ be a separable polynomial of degree $\operatorname{deg} f\geq 1$. Let $\mu$ denote the Möbius function, let $\rho_f(m)$ denote the number of residue classes modulo $m$ satisfying $f(t)\equiv 0\pmod m$, and let $\operatorname{li}(x)$ denote the logarithmic integral. Prime-input squarefree-value conjecture.

$$\sum_{p\leq x}\mu^2(f(p))=\operatorname{li}(x)\prod_{q\geq 2}\left(1-\frac{\rho_f(q^2)}{q^2}\right)+o(\operatorname{li}(x)).$$

This is the related conjecture over prime inputs mentioned in the paper. The supplied status evidence describes the quartic case $f(t)=t^4+2$ as an open problem, so the conjecture remains open in general.