Backbone.localStorage seems to replace the ajaxsync with syncing to localStorage. This library looks like it is trying to keep track of objects in local storage without removing the ajaxsync functionality.
This library caches data in localStorage and sync them with server (when online). Main difference is sync-algorithm which works with server using REST API. You can read about it here: https://github.com/Ask11/backbone.offline#how-it-works
Glad somebody did that. At some point this was a goal for Backbone.localStorage, but I ended up working with other technologies and not being willing to make time for this.