Conjecture on Fatou components of the symmetric critically finite maps

Let $g$ be the critically finite map under consideration on ${\mathbf{CP}^{n-1}}$, let $F_g$ denote its Fatou set, and let $p_m$-points denote the fixed critical points introduced in the paper. For $n\geq 5$, the Fatou set consists of the basins of attraction of the $p_m$-points.

Fatou-set conjecture. For $n\geq 5$, $F_g$ consists of the basins of attraction of the $p_m$-points.

The corresponding assertion is proved in the paper for $n=3,4$, while the general case is stated to remain open.