I'm going to go against the norm here and predict yes. There are 2 reasons the PS3 lost its BC with the PS2. One, having PS2 emulation for the PS3, especially HW emulation, added even more cost to the already expensive PS3. Sony got rid of it to minimize losses at the beginning and to maximize profit later on. And two, for years after the PS3 launched, the PS2 was still selling extremely well, and at a profit to boot. If Sony had kept BC in the PS3, this would have cut into PS2's profitable sales, sales the company needed.
However, with the PS4, not only will the PS2 be stopping production, but any HW they would need to add from the PS2 to the PS4 to get 95%-100% accurate emulation would cost them hardly anything. And while BC may not be a HUGE draw, it still is a nice addition. Especially for older gamers who wish to play their old favorites, but not want 3-4 systems clogging up their entertainment stand.
@ HappySqurriel
Yes, it would cost them some time and money to develop an emulator for the PS4. But, how do you think they are doing it on the PS3? With an emulator. Those PS2 games you're downloading aren't just an ISO of the original game. Included with them is a stand alone emulator for each game. While I'm sure it starts off with the same base emulator code, they still have to tweak it for each game (kind of like GOG does with DosBox for its games). Now, just think how much time and money they could save, as well as how many more PS2 game we could get on the PS Store more quickly, if they just had one giant emulator already included in the PS4. All they would have to do is put the games actual ISO on the PS Store and all should work fine.
