Do the email widgets follow the home screen online-offline state?

The email widget on the home screen doesn't follow the home screen online - offline state. The email clients may still connect to mail server even though the home screen has been switched to offline state with command Options > Content to offline mode (or Widgets to offline mode)  .

The email widgets behave in a different way than the other home screen web run-time widgets which don't normally establish network connections while the home screen is in offline state.