This is a follow-up to this question.
Let $\mathcal{O}$ be a Dedekind domain and $f=X^n+a_{n-1}X^{n-1}+\dots+a_1X+a_0 \in \mathcal{O}[X]$ be a $\mathfrak{p}$-Eisenstein polynomial; that is, $\mathfrak{p}$ is a prime ideal of $\mathcal{O}$ such that $a_i\in \mathfrak{p}$ for all $i$ and $a_0\not\in \mathfrak{p}^2$.
$f$ is irreducible in $\mathcal{O}[X]$ by the Eisenstein criterion. Is it irreducible in $K[X]$, where $K$ is the field of fractions of $\mathcal{O}$?