The Mandelbrot set local connectivity conjecture

The Mandelbrot set $\mathcal{M}$ is the connectedness locus of the quadratic family $p_c(z)=z^2+c$. MLC conjecture. The Mandelbrot set $\mathcal{M}$ is locally connected. Local connectivity is a central rigidity problem in complex dynamics; the claim is established for important parameter families, but remains open in general.