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on Zero area of the Multibrot set boundary

The statement

For each natural number $n$ , the multibrot set of power $n$ is defined as the set of all complex parameters $c \in \mathbb{C}$ for which the orbit of $0$ under iteration of the map $z \mapsto z^n + c$ does not escape to infinity; equivalently, it is the set of all $c \in \mathbb{C}$ such that $\|(z \mapsto z^n + c)^{[k]}(0)\| \leq 2^{(n-1)^{-1}}$ for every iterate $k \in \mathbb{N}$ , where $f^{[k]}$ denotes the $k$ -fold composition of $f$ with itself. Prove that for every natural number $n$ , the boundary of the multibrot set of power $n$ (that is, the set-theoretic frontier of this compact subset of $\mathbb{C}$ ) has Lebesgue measure zero.

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.