This is an argument that goes round and round, and regardless of my personal feelings on it, there won't be a solution to it any time soon. Nobody in MOST places is going to stop you. There are municipalities such as Laguna Beach, Ca. where it's against the law. 'nuff said there.
What is meant by a recognised class by most standards is that MOST agencies do not recognise nor offer the class. If you vacation at a PADI facility someplace in the world and flash that card thinking you're going to solo dive, chances are you're going to be disappointed. I'm well aware of the class, owning one of the VERY early TDI/SDI instructor numbers (a 3 digit one). I've also earned the ratings of Course Director for another agency and Master Instructor for a third, so I'm pretty well aware of their stances as well at this time. There are reasons for and against that both sides see as valid, so it boils down to your choice.
Again, no matter what my personal choice on the topic is, I can tell you this without a doubt. And it comes from 30+ years of experience in this sport. Regardless of what level diver you think you are, regardless of your ability to manage a crisis ABOVE the water, regardless of every single possible bit of pre-planning you may think you've covered, it only takes ONE situation underwater that a buddy may or may not have been able to assist with, for the solo diver to be PERMANANTLY dead. If you are willing to take that risk, you may dive many years and laugh it all off. Great. It's your choice. But this sport sees divers pass every year, and just as many of them are some of the most experienced our sport has to show, as are new ones.
It simply is not worth the risk for 99.9% of the divers enjoying our sport.