Conversions on android keeps a websocket connection open to receive messages. It’s supposed to generate very little overhead in battery consumption. The unified push app on android uses the same technique to provide other apps with pushes. So you could set conversations to be the push provider on your phone and would not need the unified push app any more. This way you would be using the same amount of battery as before.
The conversations xmpp support channel is very helpful if you have further questions: someone there will also very likely know if any apps with unified push support exist.