Zero area of the Multibrot set boundary

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.