Not saying a spin was definitely the issue but from what I can read, fully developed spin recovery for DA 40 (not approved for spinning that i can see) is idle, forward stick and opposite rudder. Pretty instinctive for most of us probably but perhaps not someone that has mostly flown non spin approved aircraft? Be easy to keep back stick if in a bit of a panic with the nose down.
Anyway very tragic and hopefully the cause is found so that it may help others avoid the same.