Does the introduction of Schnorr signatures require segwit? What is the relationship of segwit and Schnorr signatures? Is segwit a prerequisit for Schnorr signatures?

I've read that the transaction versioning of Segregated Witness would allow to just assign Schnorr transactions a new version number. Is that a requirement?
The current proposal for Schnorr Signatures relies on script versioning, a feature which would be introduced by SegWit. If SegWit doesn't activate, many things are still possible in bitcoin via a hard fork or a complicated soft fork, and Schnorr might be one of them. No reasonable proposals towards that end exists today.