Borel-series zero conjecture

Let $\{a_i\}_{i\in\mathbb{N}}$ be positive with finite sum, and let $\{z_i\}_{i\in\mathbb{N}}$ be distinct complex numbers such that

$$|z_i|<\rho\quad\text{for all }i,\qquad z_i\to\xi,\qquad |\xi|=\rho,$$

for some $\rho>0$. Borel-series zero conjecture. The meromorphic function

$$f(z)=\sum_{i=1}^{\infty}\frac{a_i}{z-z_i}$$

has infinitely many zeros in $\mathbb{D}(\rho)=\{z\in\mathbb{C}:|z|<\rho\}$. This is the bounded positive-coefficient analogue of the Clunie–Eremenko–Rossi phenomenon for Borel series, and the paper notes that the assumptions ensure meromorphicity in the disk; the conjecture remains open in the supplied text.