Finiteness conjecture for equilibrium points of same-sign charges

Let a finite set of point charges of the same sign be placed in $\mathbb{R}^n$, and call a point an equilibrium point if the corresponding electrostatic field vanishes there.

Same-sign equilibrium finiteness conjecture. For any such set of charges, the set of equilibrium points is finite.

The claim is presented as a folklore question and is not settled in the source. Charges of different signs can instead have a positive-dimensional equilibrium set.