Nobody. This is the principal unsolved problem in the area, which is called MLC (That the Mandelbrot set is locally connected). Two Fields medals were awarded for partial progress in this problem.
About Julia sets, some of them are locally connected, others are not. See
The deep significance of the question of the Mandelbrot set's local connectedness?
for more detail.
Remarks. All proofs in the Orsay notes are due to Douady and Hubbard, unless stated otherwise. That Julia sets corresponding to the interior of the Mandelbrot set are locally connected is also not known. It is only known for parts of this interior: for the main hyperbolic component this was proved by Fatou, and for the rest of hyperbolic components by Douady-Hubbard. But the existence of components of the interior, other than hyperbolic components is not known.