Linear slice conjecture for electrostatic potentials

Let $A_1,\ldots,A_n\in\mathbb{R}^3$ be charge locations, and let

$$V\colon\mathbb{R}^3\setminus\{A_1,A_2,\ldots,A_n\}\to\mathbb{R}$$

be the electrostatic potential defined in the source. An equilibrium of a restriction is a critical point of that restriction.

Linear slice conjecture. The restriction of $V$ to any straight line or flat plane in $\mathbb{R}^3$ has at most $O(n)$ equilibria.

This conjecture seeks a linear bound for one- and two-dimensional slices, extending the known linear-cell behavior for certain one-dimensional tessellations. The supplied text gives no resolution.