Failed items (for example when the publication failed) appear in the human task list under failed, but it is not visible in the trinity icon.
Even when you switch between other apps and refresh the screen.

I think this could be very confusing - and in fact I fell into the trap - that publishers believe it all went well because there was no little number showing up next to the pulse trinity icon.

From a user perspective, it is a new item. It is a known process, but it is new for me that it failed and that needs my attention again.