Borcea's conjecture on zeros in a disk

Let $0<\rho<\infty$ and $\theta\in\mathbb R$. Define

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

where $z_k\in\mathbb C$, $|z_k|<\rho$, $z_k\to\rho e^{i\theta}$, $a_k>0$, and

$$\sum_{k=1}^{\infty}a_k<\infty.$$

Borcea's conjecture. The function $f$ has infinitely many zeros in $|z|<\rho$.

This is a disk analogue of the planar equilibrium-point conjecture and has the interpretation of finding equilibrium points for the associated logarithmic potential. The source presents it as an analogue; no resolution is supplied here.