Complex-charge equilibrium zero conjecture

Let $\{a_n\}_{n\in\mathbb{N}}$ and $\{z_n\}_{n\in\mathbb{N}}$ be complex sequences with distinct $z_n$ satisfying

$$a_n\overline{z_n}>0,\qquad z_n\to\infty,\qquad \sum_{n=1}^{\infty}\left|\frac{a_n}{z_n}\right|<\infty.$$

Complex-charge equilibrium zero conjecture. The meromorphic function

$$f(z)=\sum_{n=1}^{\infty}\frac{a_n}{z-z_n}$$

has infinitely many zeros. The paper proposes this as an analogue of the Clunie–Eremenko–Rossi conjecture for certain potentials generated by complex charges; the supplied text gives no resolution.