The full wreath-product Galois group conjecture for $f_p$

Let $p$ be a prime, let $f_p$ be the polynomial studied in the source, let $g_p$ be its associated auxiliary polynomial, and write

$$h_p=\deg(g_p).$$

Let $\mathbb Q(f_p)$ denote the splitting field of $f_p$ over $\mathbb Q$. The full Galois group conjecture. The extension $\mathbb Q(f_p)/\mathbb Q$ is Galois with Galois group

$$(\mathbb Z/2\mathbb Z)^{h_p}\rtimes S_{h_p},$$

where $S_{h_p}$ acts naturally by permuting the factors of $(\mathbb Z/2\mathbb Z)^{h_p}$. This predicts the maximal Galois group compatible with the quadratic extensions arising from the roots of $f_p$; the source reports computational evidence but does not state a resolution.