General irreducibility conjecture for integer polynomials with prime-power constant term

Let $p$ and $q$ be prime numbers, let $u\geqslant 1$, and let

$$f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{q}x^{q}+p^{u}\in\mathbb{Z}[x].$$

Irreducibility conjecture. If $p\nmid a_{n}a_{q}$, $q\nmid u$, and

$$p^{u}>|a_{n}|+|a_{n-1}|+\cdots+|a_{q}|,$$

then $f(x)$ is irreducible over $\mathbb{Q}$. This conjecture seeks a common irreducibility criterion extending the preceding results for the cases $q=1$, $q=2$, and $q=3$; its resolution is not given here.