It appears that some process or background process is invoking the same API as HP HPCustParticipation task was doing, before I changed the option to "allow it to run only when the user is loggued on".
It could be a process or a Task. I would go across the task, and put this option for whatever task that seems suitable to switch to this option.
Also in my case I tried and suggested some steps found elsewhere, which implied to play with user rights and deleting some files related to password retention. One of them had a serious effect about making all password record by Edge disappear. But it only
worked a few days.
Enough bizarre is the fact that when I changed to HP task option, and rebooted, all passwords disappeared in Edge where restored. As said by many, this is a serious bug in the crypto API. From then all problems were gone.
What is process or task that trigger the but through some bug in the API? It could be HP tasks or other task that is set to start automatically, either through the Task Scheduler, or may by an auto-start job. I would try to investigate which of them, by
suspending or disabling them.