Boundedness 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.

Bounded-zero-set conjecture. The zero set $Z(F)$ is bounded.

If true, this would rule out common zeros at infinity. The source presents the claim as a conjecture and does not give a resolution.