For an irreducible polynomial $f \in \mathbb{Z}[x]$ with $f(n) \ge 1$ for sufficiently large $n$, does there exists a constant $c = c(f) > 0$ such that $\sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x$?

For an irreducible polynomial $f \in \mathbb{Z}[x]$ with $f(n) \ge 1$ for sufficiently large $n$, does there exists a constant $c = c(f) > 0$ such that $\sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x$?

Note that it is unclear whether the polynomial should have integer coefficients or merely be integer-valued. We assume the former.