Sony is working on PS2 backward compatibility for PlayStation 4 - GSMArena blog
GitHub - xXxTheDarkprogramerxXx/Ps2-Classics-emulators-for-Ps4: Ps2 Classics emulators for Ps4, this will allow you to use many different versions of emulator, which could improve or decrease the compatibility
As PS2 Turns 20, These Classics Must Come to PS4 - Feature | Push Square