I have tried quite a bit (I've got quite a bit of JP PS2 games) and all got the same issue, just to mention some:
Ape Escape 2
Gran Turismo 3 and 4
The Three FMA Games
Among others, I don't think it's the games itself since after the PS3 gets to certain temp, they do launch and run flawlessly...
I recently bought a CECHA00 PS3 from Japan for relatively cheap, it works all good so far except for the fact that most of the time, it won't play PS2 games, I just get a black screen.
I say "most of the time" because it seems to play PS2 games but just under very specific conditions (I still...