Suppose $\{p_k\}$ is a sequence of polynomials with $p_k(0)=1$. Let $a_1,a_2,\ldots$ be an enumeration of all of the zeros of the $p_k$. Suppose that $$\prod_{k=1}^\infty p_k(z)$$ converges uniformly on compact subsets of $\mathbb{C}$. Is there some permutation $\sigma:\mathbb{N}\to\mathbb{N}$ such that $$f_\sigma (z)=\prod_{k=1}^\infty \left(1-\frac{z}{a_{\sigma(k)}}\right)$$ converges uniformly on compact subsets of $\mathbb{C}$?
This is a follow-up to Factoring a convergent infinite product of polynomials., in which an example of such $\{p_k\}$ is given along with a permutation $\sigma$ for which the product $f_\sigma (z)$ does $\underline{\text{not converge}}$.