A client of mine went there. She did lose weight while on it, a lot of weight pretty rapidly and gained it back super fast the second she quit and starting eating better.
She followed it religously, and nearly passed out more than once during our sessions. From what they allowed her to eat, it couldn't have been one bite over 1200 calories, and believe me, she needed way more than that. I finally had to tell her that she would have to get more food in on her workout days or she wouldn't be able to safely workout. When she voiced her concern with LA Weightloss over not having enough food for exercise - their official plan was to add 1 veggie on the days she exercised.
Another client of mine went as well, and the same thing was happening to her while exercising.
When you meet your goal and get to maintenance, you are allowed 1 or 2 more servings (I believe) of a few things. Still well below a healthy amount.
So, in a nutshell, yes, you most certainly will lose weight while on the LA Weightloss plan, and keep it off, if and only if you are willing to adhere to a very stringent extreme caloric deficit forever.