Well this thing just got me puzzled. It does say on the official site that cards under R7870 are not supported, but that maybe just due to theoretical lack of power - not software conditions that would absolutely prevent the game from working. I'm pretty sure that my system will run it on some level of specs, I'm not an FPS freak, so as long as it does't run on 20 it's all good...
I'm sure you could run the game.
I have a Sapphire 7850 OC 2GB card collecting dust at the moment. I'm sure it would be capable of running the Witcher 3 at low settings.
They have to list something as the requirement. Although, it doesn't mean that an overclocked card that is technically below spec., can't run it.
You just need to be happy enough with low settings and low'ish fps.
I'm speculating of course, but we've seen this before. It's a matter of what you're willing to live with, to play the game.