Should you include your main supervisor as a co-author?
ASK THEM.
Academia is a very 'political' and 'incestuous' environment.  If it is common to do that, then in order for you to succeed there, you might should do what they do.
  si fueris Rōmae, Rōmānō vīvitō mōre; si fueris alibī, vīvitō sicut ibi ‎(“if you should be in Rome, live in the Roman manner; if you should be elsewhere, live as they do there”); which is attributed to St Ambrose.

If you don't, and they expect it, then you might just piss them off.  They may never mention it, but they might take it out on you in ways you could never prove. Retaliation likely won't be something you even see - not getting that scholarship, journal, presentation, etc.

TL;DR - Ask them what to do, then do it.