The polynomial Erdős–Wagstaff density conjecture

Let $P(x)$ be a polynomial, and let $\bar{\mathbf{d}}$ denote upper density. For a positive integer $T$, consider the set of integers $n$ for which $p-1$ divides $P(n)$ for some prime $p>T$. Polynomial Erdős–Wagstaff conjecture. For all $\epsilon>0$ and all polynomials $P(x)$, there exists a $T=T(\epsilon,P)$ such that

$$\bar{\mathbf{d}}\bigl(\{n: p-1\mid P(n)\text{ for some prime }p>T\}\bigr)<\epsilon.$$

This is proposed as the polynomial analogue of the Erdős–Wagstaff result used to prove that the range of Euler's $\varphi$-function has density zero. The statement is presented as a needed variant for extending that argument to polynomial values and is not resolved in the supplied text.