Hyperoctahedral Galois group conjecture for sorted binomial polynomials

Let $P_n(z)$ be the sorted binomial polynomial. For odd $n=2m+1$, the source identifies the relevant target group as the $m$th hyperoctahedral group. Hyperoctahedral Galois group conjecture. $P_n(z)$ has full Galois group for even $n$. For $n=2m+1$, the Galois group of $P_n(z)$ is the $m$th hyperoctahedral group. The preceding discussion records that the odd case is related to the Galois group of $q_n(z^2)$ and that broad full-Galois-group results are known only under additional hypotheses; the displayed assertion itself is not resolved in the supplied text.