I have the beta android app 5.1.5-34, and a Huawei Nova with android 6.0. The app Doit it is on the protected apps on the battery settings of my Huawei (so it have permission to be and sync on background)

But when i do not open the app for many time, and add a task on the web version, it do not appear on the android version until I sync manually opening the app. And sometimes in the day i see my widget list clean, when i do have tasks on the android app, and that tasks appear on the widget when i sync manually, but after a time it go back to clean.