We currently do not use the branch status at all. But we probably should. At least in the case where there are no known tasks at all. Currently a "timeout" branch status is not correctly reflected. From the discord:

@JelteF #4677 is showing as in progress on the cf app, but cfbot gave up on it and set the branch to "timeout". looks like cirrus/github flaked out on us with "FAILED_PRECONDITION: GitHub API failure!!". that's not our fault but i realise now that in the place in cfbot_util.py where the gc() function whacks b0rked branches with "timeout", it probably needs to dump a push-branch-status job into work_queue to tell the cf. don't have time to work on it right now and it'll clear itself on the cf page soon enough when it is re-scheduled for a bitrot check (no quadratic backoff for timed out branches, only failed tasks), so just noting the observation here...
Thomas Munro — Today at 02:10
additional clue: i manually did
insert into work_queue (type, key, status) values ('post-branch-status', '188732', 'NEW');
and it posted the notification, but it didn't change the display so perhaps the cf app doesn't understand status "timeout" yet