Why does diode keep conducting even after voltage across it is negative
The inductor's stored magnetic energy will try and keep current flowing through both itself and D2 by generating sufficient back-emf to forward bias D2. That's pretty much what an inductor will do in this situation.

Why does diode keep conducting even after voltage across it is
negative

You might think that it's negative but, until the inductor has pretty-much got rid of most of its stored energy, the diode will be kept forward biased (like it or not).