Irreducibility conjecture for the polynomials $C_{X,\lambda_2}$

Let $C_{X,\lambda_2}$ denote the polynomial discussed in the concluding remarks, with coefficients in ${\mathbb Q}$. Irreducibility conjecture. For even $\lambda_2\ge2$, $C_{X,\lambda_2}$ is irreducible over ${\mathbb Q}$. For odd $\lambda_2\ge3$, $C_{X,\lambda_2}$ is the product of $X-\lambda_2$ and an irreducible polynomial over ${\mathbb Q}$.

The paper verifies the claim computationally for even $\lambda_2\le600$ and discusses related irreducibility checks, but no general proof is given.