Uniform quasicircle conjecture for CaTherine wheels

Let $R>r>0$, let $G$ be a Kleinian group, and let $f$ be a CaTherine wheel invariant under $G$. Assume that the injectivity radius of $\bbH^3/G$ lies everywhere in the interval $[r,R]$. Uniform quasicircle conjecture. There is a finite constant $K(r,R)$, depending only on $r$ and $R$, such that $f$ is a $K(r,R)$-CaTherine wheel. This would provide uniform geometric control of CaTherine wheels under a two-sided injectivity-radius bound and may help estimate the conformal dimension of boundaries of hyperbolic groups.