Finiteness of the zero set of a two-dimensional finite electrical field

Let $F=(X,Y)$ be a two-dimensional electrical field generated by finitely many point charges, and let

$$Z(F)=\{(x,y)\in\mathbb R^2:F(x,y)=(0,0)\}$$

be its zero set.

Finite-zero-set conjecture. The zero set of a two-dimensional finite electrical field consists of only finitely many points.

This is the global finiteness conclusion the paper seeks for common zeros of the two field components. The source presents it as a conjecture and does not provide a resolution.