Family of irreducible polynomials over $\mathbb{Z}$

Question

Note. In this question I am using the following definition of irreducible polynomial: "Polynomial is irreducible if it cannot be factored into polynomials with coefficients in the same domain that have both a positive degree." (See this Wikipedia article)

I am trying to prove the following claim:

Let $P_n(x)=D_n(x,1)-x^n$ , where $D_n(x,1)$ is a Dickson polynomial of the first kind with parameter $\alpha=1$ . Then, $P_n(x)$ is irreducible over $\mathbb{Z}$ for all even $n$ greater than two.

I have verified this claim for all $n$ up to $2000$. Here is the link to the SageMathCell.

Some of my observations:

1. In case when $n$ is even $P_n(x)$ is even , i.e. $P_n(-x)=P_n(x)$.
2. In case when $n=2p$, where $p$ is a prime number, vector of the slopes of the Newton polygon of the polynomial $P_n(x)$ with respect to $p$ is of the form: $[-1/(2p-2),-1/(2p-2),\ldots,-1/(2p-2)]$ .
3. In case when $n$ is even Eisenstein's criterion, Cohn's irreducibility criterion and Perron's irreducibility criterion cannot be applied to polynomial $P_n(x)$ .

Any hints are welcomed.