If considering a hollow conducting sphere with a surrounding uniform charge distribution, for example, it will have a constant and uniform potential throughout the inside of the hollow sphere because $\phi \sim 1/r$. But if instead there were dipoles, quadrupoles, octupoles, etc. uniformly surrounding a hollow sphere with $\phi \sim 1/r^n$ and $n$ an arbitrary integer, is the potential inside the sphere necessarily uniform everywhere?
The basic answer appears to be yes by Gauss's Law since there is no charge inside the hollow sphere. And I've seen geometric arguments for the $n = 1$ case, but are there any general proofs for arbitrary $n$?