Linear bound conjecture for weighted Euclidean-distance equilibria in the plane

Let $A_1,\ldots,A_n\in\mathbb{R}^2$ be points with positive real weights, and consider the associated weighted Euclidean distance function. Its equilibria are the critical points of this function.

Linear weighted-distance conjecture. The number of equilibria of the weighted Euclidean distance function defined by $n$ points with positive real weights in $\mathbb{R}^2$ is at most some constant times $n$.

The conjecture contrasts the potentially quadratic number of cells in planar weighted Voronoi tessellations with the conjectured linear number of equilibria. The supplied text gives no resolution.