Finite self-similarity conjecture for Julia sets of rational functions

Let $f$ be a complex rational function, and let $J(f)$ denote its Julia set. A compact metrizable space is finitely self-similar if it arises from a finite self-similarity system, meaning that its self-similarity equations involve only finitely many spaces. Julia-set finite self-similarity conjecture. The Julia set $J(f)$ of any complex rational function $f$ is finitely self-similar.

This is stated as a more tentative conjecture after the proposed finite self-similarity of finite simplicial complexes. The source gives no resolution.