One-dimensional Maxwell bound for critical points of point-charge potentials

Consider an $l$-tuple of points $(x_1,y_1),\ldots,(x_l,y_l)$ in $\mathbb{R}^2$, charges $(\zeta_1,\ldots,\zeta_l)$, and $\alpha\ge \frac{1}{2}$. Define

$$V_{\alpha}^*(x)=\sum_{i=1}^l\frac{\zeta_i}{\bigl((x-x_i)^2+y_i^2\bigr)^{\alpha}}.$$

One-dimensional Maxwell bound. For any values of the charges, the function $V_{\alpha}^*(x)$ has at most $2l-1$ real critical points. This is a stronger one-dimensional version of the relative Maxwell inequality, supported by numerical evidence and stated in the source as still open.