What ShinkanTrain said. The last a read about it, the PS2 only switches into PS1 mode on a trigger from the optical drive subsystem, and then most of the memory and other hardware used to run homebrew is deactivated. AFAIK no-one’s yet found a way to trigger the change in software and keep the connection to wherever you’re loading your game from.

I believe that on certain revisions of the console, MechaPwn can overcome the protection, but you still need a “Playstation 1” CD in the drive to actually run something, as ShinkanTrain wrote.