Conjecture on self-similarity of non-escaping loci

Let $d_0$ and $d_\infty$ be the degree parameters of the family $\\{F_c\\}_{c\in\mathbb{C}^*}$, and let

\mathcal{M}_{d_0,d_\infty}=\left\\{c\in\mathbb{C}^*: F_c^n(1)\not\to 0\text{ and }F_c^n(1)\not\to\infty\right\\}

be its non-escaping locus. For a stationary type irrational number $\theta=[0;N,N,N,\ldots]$, write $c(\theta)$ for the corresponding parameter.

Self-similarity conjecture. For every stationary type irrational number $\theta=[0;N,N,N,\ldots]$, the non-escaping locus $\mathcal{M}_{d_0,d_\infty}$ is asymptotically self-similar at $c(\theta)$, and there is a hyperbolic renormalization operator associated to it.

This is presented as preliminary evidence for asymptotic self-similarity, with the expected mechanism being hyperbolicity of an appropriate renormalization operator. The source gives no resolution of the assertion.