Linear bound for equilibria of weighted Euclidean distance functions in the plane

Prove that the number of equilibria of the weighted Euclidean distance function defined by n points with positive real weights in R^2 is at most a constant multiple of n.

Background

For positive but unequal charges, the limiting function is a multiplicatively weighted Euclidean distance function. Unlike the unweighted case, its planar Voronoi tessellation can have quadratically many cells, and there is no established Delaunay-type dual structure.

Despite this combinatorial complexity, the authors conjecture that the number of equilibria remains linear in the number of weighted points.

References

We think not. \begin{conjecture} The number of equilibria of the weighted Euclidean distance function defined by $n$ points with positive real weights in $R2$ is at most some constant times $n$. \end{conjecture}

— Counting Equilibria of the Electrostatic Potential  (2501.05315 - Edelsbrunner et al., 9 Jan 2025) in Conjecture 2D_weighted_Euclidean_distance, Section 4.2