Simple unsolved math problem, 6

Probably everyone has at least seen the Mandelbrot set in some form, as it’s a popular object of mathematical artists. Here’s a picture from Wikipedia:

The formal definition is as follows. Let $f_c (z)=z^2+c$, where $c\in \mathbb{C}$ is a complex number. The Mandelbrot set $X$ is the complex plot of the set of complex numbers $c$ for which the sequence of iterates $latex f_c(0)$, $latex f_c (f_c (0))$, $latex f_c (f_c (f_c (0)))$, etc., remains bounded in absolute value.

Conjecture: The Mandelbrot set $X$ is locally connected.

We say $X$ is locally connected if every point $x\in X$ admits a neighborhood basis consisting entirely of open, connected sets. If you know a little point-set topology, this is a conjecture whose statement is relatively simple.