Convergence of finite-root Julia sets to template Julia sets

Fix a parameter pair $(c_0,c_1)\in\mathbb{C}^2$ and let ${\bf s}$ be a template. For each $n$, let $\overline{\bf s}^{n}$ denote the finite $n$-root of ${\bf s}$, and let $J_{c_0,c_1}(\overline{\bf s}^{n})$ and $J_{c_0,c_1}({\bf s})$ be the associated Julia sets. Finite-root convergence conjecture. For every $\delta>0$, there exists $N\in\mathbb{N}$ such that, for all $n\geq N$,

$$d\bigl(J_{c_0,c_1}(\overline{\bf s}^{n}),J_{c_0,c_1}({\bf s})\bigr)<\delta.$$

The claim formalises convergence suggested by the numerical study of templates sharing longer initial segments. The source gives no evidence of a proof or disproof.