Irreducibility conjecture for even sorted binomial polynomials

Let $P_n(z)$ be the sorted binomial polynomial with parameter $n$. Irreducibility conjecture. $P_n(z)$ is irreducible over $\mathbb{Q}$ for all even $n$. The analogous factorization for odd $n$ has just been proved in the surrounding text, while this even-$n$ assertion is supported only by numerical computations for low $n$.