Maxwell's quadratic upper-bound conjecture for electrostatic equilibria

Let $A_1,\ldots,A_n$ be point charges in $\mathbb{R}^3$, and let the electrostatic potential be generated by these charges. An equilibrium is a zero of its electric field.

Maxwell's conjecture. The electrostatic potential defined by $n$ point charges in $\mathbb{R}^3$ has at most $(n-1)^2$ equilibria.

This is the 1873 conjecture motivating the paper. The source states that the bound is attained for $n=3$, while only substantially larger general upper bounds are known.