Distinct asymptotic directions for the components of a finite point-charge electrical field

Let $F=(X,Y)$ be the electrical field generated by finitely many point charges in the plane. In the Type I and Type II domains, consider the asymptotic directions of the zero sets $X=0$ and $Y=0$.

Distinct-asymptotic-directions conjecture. The asymptotic directions of the points in the zero sets $X=0$ and $Y=0$, in the Type I and Type II domains, are distinct. Hence, outside some large disk, there are no common zeros in either domain for $X$ and $Y$; equivalently, $F$ has no zeros far from the origin.

This conjecture would establish the absence of common zeros at infinity for the two components of the field. The source does not provide a resolution.