Conjecture on zeros of meromorphic logarithmic potentials in the plane

Let

$$f(z)=\sum_{k=1}^{\infty}\frac{a_k}{z-z_k},$$

where $a_k>0$, $z_k\in\mathbb C$, $z_k\to\infty$, and

$$\sum_{z_k\ne0}\left|\frac{a_k}{z_k}\right|<\infty.$$

Conjecture on infinitely many zeros. The function $f$ has infinitely many zeros in $\mathbb C$.

The assumptions make $f$ meromorphic in the plane and identify it with the complex conjugate of the gradient of an associated subharmonic logarithmic potential. The claim has an electrostatic interpretation and is known under additional hypotheses such as $\sum_{|z_k|\le r}a_k=o(\sqrt r)$ or $\inf\{a_k\}>0$, but is not established in general.