This is a follow-up of the question Dividing a spherical cap into three equal wedges where the $n=3$ case was shown.

Motivation: Optimal ways to cut an orange.

In this problem, we have a spherical orange of radius $R$, and we do not wish to eat its central column which is modelled as a cylinder of radius $r>0$. Part of the procedure involves an initial cut down the orange tangential to the central column, which produces a spherical cap, then further splitting the cap into $n$ wedges where each cut passes through the midpoint of the initial cut. The goal is to make these $n$ wedges have equal volumes.

The question is whether each of these $n$ wedges can have the same angle. A negative answer has been shown for the $n=3$ case, by deriving closed-form expressions for each volume and proving that they are incompatible with $\alpha_1=\alpha_2=\alpha_3=\pi/3$ and $V_1+V_2+V_3=\pi(R-r)^2(2R+r)/3$, which is the volume of the whole spherical cap.

Can this result be extended to $n$ wedges of identical volume with angle $\pi/n$?