Connectedness characterisation of template Mandelbrot sets

Let ${\bf s} \in {\cal L}(\{0,1\})$ be a symbolic template, and let $(c_0,c_1) \in \mathbb{C}^2$. The fixed-template Mandelbrot set is

$${\cal M}_{\bf s}=\{(c_0,c_1)\in\mathbb{C}^2\;\slash\;o_{\bf s}(0)\text{ is bounded}\},$$

and the fixed-map Mandelbrot set is

$${\cal M}_{\bf c_0,c_1}=\{{\bf s}\in{\cal L}(\{0,1\})\;\slash\;o_{\bf s}(0)\text{ is bounded}\}.$$

Here $J_{c_0,c_1}({\bf s})$ denotes the corresponding template Julia set. Connectedness conjecture. For every template ${\bf s}$,

$${\cal M}_{\bf s}=\{(c_0,c_1)\in\mathbb{C}^2\;\slash\;J_{c_0,c_1}({\bf s})\text{ is connected}\},$$

and, for every $(c_0,c_1)\in\mathbb{C}^2$,

$${\cal M}_{\bf c_0,c_1}=\{{\bf s}\in{\cal L}(\{0,1\})\;\slash\;J_{c_0,c_1}({\bf s})\text{ is connected}\}.$$

This proposes the analogue of the classical relation between the Mandelbrot set and connectedness of Julia sets for template iterations. The source presents it as a question following discussion of the failure of one classical implication, and gives no resolution.