Connectedness conjecture for the multi-Mandelbrot set

Let $\mathbb{D}(2)$ be the disc of radius $2$ centered at the origin, and let $\phi_{c_0,c_1}$ denote the template-iteration map associated with the parameter pair $(c_0,c_1)$. The multi-Mandelbrot set is

$${\cal MM} = \{ (c_0,c_1) \in \mathbb{D}^2(2),\ \text{such that } \phi_{c_0,c_1}(1) =1 \}.$$

Multi-Mandelbrot connectedness conjecture. The set ${\cal MM}$ is a connected set in $\mathbb{C}^4$.

The set ${\cal MM}$ is described as the set of all infinitely well-behaved parameter pairs in $\mathbb{D}(2) \times \mathbb{D}(2)$. The paper notes that three-dimensional slices of the finite approximations need not be connected, while leaving the connectedness of the full set unresolved.