Gabrielov–Novikov–Shapiro monotonicity conjecture for generalized electrostatic potentials

Let $A_1,\ldots,A_n$ be point charges in generic position, and for $p\geq 1$ define

$$V_p(x)=\sum_{i=1}^n\frac{\zeta_i^p}{\lVert x-A_i\rVert^p}.$$

For $j\in\{0,1,2,3\}$, let $\#_j$ denote the limiting number of index-$j$ equilibria of $V_p$ for sufficiently large $p$.

Gabrielov–Novikov–Shapiro conjecture. For $n$ unit point charges in generic position, and all $p\geq 1$, the number of index-$j$ equilibria of $V_p$ is at most $\#_j$.

The conjecture expresses the expected non-decrease of the number of equilibria with $p$. The paper states that it provides a counterexample, so this conjecture is refuted.