Gabrielov–Novikov–Shapiro line-restriction conjecture for generalized electrostatic potentials

Let $V_p$ be the generalized electrostatic potential associated with $n$ point charges in $\mathbb{R}^3$, with $p\geq 1$. Restrict $V_p$ to a straight line, and call a critical point of this restriction an equilibrium.

Gabrielov–Novikov–Shapiro conjecture. For all $p\geq 1$, the number of equilibria of the restriction of $V_p$ to any straight line in $\mathbb{R}^3$ is at most $2n-1$.

The conjecture strengthens an elementary linear bound available for even $p$. The supplied text gives no resolution of this stronger statement.