Maximal area conjecture for filled Julia sets

Let $c$ be a parameter in the Mandelbrot set, and let $K_c$ denote the associated filled Julia set. The unit disk is the filled Julia set of maximal area.

Maximal area conjecture. The maximal area of $K_c$ is

$$\pi,$$

attained by the unit disk.

This conjecture concerns the geometric size of filled Julia sets across the Mandelbrot set and is supported by the experimental evidence described in the paper; its resolution is not specified here.