Parameter self-similarity conjecture for critical quasicircle maps

Let $d_0$ and $d_\infty$ be the local degrees of the critical fixed points at $0$ and $\infty$, respectively, and let $\{F_c\}_{c\in\mathbb{C}^*}$ be the one-parameter family of rational maps of degree $d_0+d_\infty-1$ with those critical fixed points and a full-degree critical point at $1$ satisfying $F_c(1)=c$. For each $\theta\in\operatorname{IrratPer}$, let $c(\theta)$ be the parameter for which $F_{c(\theta)}$ admits an invariant quasicircle through $1$ with rotation number $\theta$. Parameter self-similarity conjecture. For every $\theta\in\operatorname{IrratPer}$, the bifurcation locus of $\{F_c\}_{c\in\mathbb{C}^*}$ is asymptotically self-similar at $c(\theta)$, with a universal self-similarity factor depending only on $(d_0,d_\infty,\theta)$. Computer pictures suggest this phenomenon, but no general proof is given; it remains open.