I have been actively glancing over to the settings page to see whether “Offline” has been activated for my account. And it was there so I decided to activate it:
The settings for Offline is displayed, but not changeable (except to enable or disable the feature):
I have no idea how the algorithm selects which labels to always synchronize.
Now I have enabled it and synced the emails:
When offline, this indicator is seen:
When using it offline I don’t really notice the difference, except when my Firefox gave me hiccups as it try to connect to the Internet for other websites.