Azure Scheduler stops triggering Azure WebJob

Few days ago I noticed that our reminder job stopped running with the following error:
Http Action - Response from host 'bornreminders.scm.azurewebsites.net': 'Unauthorized'
Response Headers: Date: Thu, 19 May 2016 09:02:01 GMT
Server: Microsoft-IIS/8.0
WWW-Authenticate: Basic realm="site"

No changes were made to the web job or to the app, so we assumed it had something to do with Azure services. This behaviour was similar to the issue here http://stackoverflow.com/questions/35082272/azure-scheduler-fails-to-trigger-ondemand-webjob-published-with-webapp, but after updating publishing NuGet to be the 1.0.11 version, the issue was not resolved.
However, we had another webjob that was continuing to be triggered by an Azure scheduler successfully, and after comparing their setting we found that lack of authentication settings was causing the issue.

To enter the authentication settings:
-> Open the Azure Scheduler in the new Azure Portal
-> Select the job that having an issue
-> Click on an Action tile
-> Authentication Settings
-> Select “Basic” as authentication type and enter user name and password used in the webjob publishing profile.

Azure Webjob authentication settings

Azure Webjob authentication settings

One thing to point out is that publishing webjob from Visual Studio will erase those settings. Not sure what needs to be done so the authentication settings are published together with the job.