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on Density of hyperbolicity for Multibrot sets

The statement

For an integer $n \geq 2$ , the Multibrot set of power $n$ , denoted $\mathcal{M}_n$ , is the set of all complex parameters $c$ for which the orbit of $0$ under iteration of the map $f_c(z) = z^n + c$ does not escape to infinity. Equivalently, $\mathcal{M}_n$ is the set of all $c \in \mathbb{C}$ such that $\|f_c^{[k]}(0)\| \leq 2^{(n-1)^{-1}}$ for all $k \geq 0$ , where $f_c^{[k]}$ denotes the $k$ -fold iterate of $f_c$ .

A point $z \in \mathbb{C}$ is said to be part of an attracting cycle of period $m$ for a holomorphic map $f: \mathbb{C} \to \mathbb{C}$ if $m > 0$ , $f^{[m]}(z) = z$ , the iterate $f^{[m]}$ is differentiable at $z$ , and $\|\operatorname{deriv}(f^{[m]})(z)\| < 1$ .

Prove that the set of parameters $c \in \mathbb{C}$ for which $f_c(z) = z^n + c$ has an attracting cycle is dense in $\mathcal{M}_n$ . That is, prove that

\mathcal{M}_n \subseteq \overline{\{c \in \mathbb{C} : \exists\, m \geq 1,\, \exists\, z \in \mathbb{C},\; z \text{ is part of an attracting cycle of period } m \text{ for } f_c\}}.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.