Let $f(x)\in \mathbb{Z}[X]$ be a polynomial of degree at least $2$. We denote the set of primes $p$ for which $f(x)$ is injective modulo $p$ as $\mathcal{T}$. Then, can we say something about the proportion of polynomials $f(x)$ for which cardinality of the set

$$\#\mathcal{T}(y)\ll \frac{y}{(\log{y})^2}.$$