Douady–Hubbard local connectivity conjecture for the Mandelbrot set

Let $p_c(z)=z^2+c$ be the quadratic family, let $\mathcal{M}$ be its Mandelbrot set, and let $\mathcal{B}=\partial\mathcal{M}$ be the quadratic bifurcation locus.

Douady–Hubbard's local connectivity conjecture. The set $\mathcal{B}$ is locally connected.

This is equivalent to local connectivity of the Mandelbrot set and is described in the source as a central open problem in holomorphic dynamics. It is important in part because it implies density of hyperbolicity for quadratic polynomials.