[thejeff]

I assume it does go inactive, because I've had the AI declare war again after a period without conflict.

There are still advantages to paying attention to the AIs declarations, even if they will attack on the same turn.
You should be prepared for an attack, but if you border 2 AIs and are already at war with one, don't provoke war with the other by attacking, since you assume you're at war with all AIs every turn.