Density of hyperbolicity conjecture for the Mandelbrot set

Recall that the Mandelbrot set is

$$\mathcal{M}:=\{c\in\mathbb{C}:0\in K_c\}.$$

For the quadratic polynomial $f_c(z)=z^2+c$, let

$$\mathcal{H}:=\{c\in\mathcal{M}:f_c(z)=z^2+c\text{ is hyperbolic}\}.$$

Density of hyperbolicity conjecture. The set $\mathcal{H}$ is dense in $\mathcal{M}$.

This is one of the most important open problems in holomorphic dynamics. It asks whether every parameter in the Mandelbrot set can be approximated by parameters for which the corresponding quadratic polynomial is hyperbolic.