Continuity of template Julia sets under common roots

Fix a parameter pair $(c_0,c_1)\in\mathbb{C}^2$ and let ${\bf s}$ be a template. For a template ${\bf t}=(t_k)$, its $k$-root is the finite sequence $\overline{\bf t}^{k}=t_1,\hdots,t_k$; two templates have a common $k$-root when they agree through their $k$-th position. Common-root continuity conjecture. For every $\delta>0$, there exists $n\in\mathbb{N}$ such that, whenever ${\bf s}_1$ and ${\bf s}_2$ have a common $n$-root,

$$d\bigl(J_{c_0,c_1}({\bf s}_1),J_{c_0,c_1}({\bf s}_2)\bigr)<\delta,$$

where $d$ is the Hausdorff distance between two sets. This conjecture is motivated by numerical simulations showing that templates agreeing for an initial segment produce increasingly similar Julia sets. No proof or resolution is supplied.