Equivalent half-plane formulation of Borcea's conjecture

Let

$$F(w)=\sum_{k=1}^{\infty}\frac{a_k w_k}{w-w_k},$$

where $w_k\in\mathbb C$, $\operatorname{Re}w_k>\frac12$, $w_k\to\infty$, $a_k>0$, and

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

Equivalent half-plane formulation. The function $F$ has infinitely many zeros in $\operatorname{Re}w>\frac12$.

The source derives this statement from the disk conjecture by the change of variables $w=(1-z)^{-1}$, $w_k=(1-z_k)^{-1}$, and $f(z)=wF(w)$. It is therefore a restatement of Borcea's conjecture rather than a separate claim.