Symmetry equality conjecture for König maps

Let $p$ be a normalized polynomial, let $n\geq 2$, and let $K_{p,n}$ be the König root-finding map. Write $\Sigma p$ for the symmetry group associated with $p$ and $\Sigma K_{p,n}$ for the corresponding symmetry group of $K_{p,n}$. Symmetry equality conjecture. For every $n\geq 2$ and every normalized polynomial $p$,

$$\Sigma K_{p,n}=\Sigma p.$$

The preceding results establish inclusion in one direction, while the source proposes equality as a conjecture; no resolution is supplied.