Amethyst should never run in the background (besides the seconds that it takes to process notifications). 

If the app is running then there is something wrong with the setup. I have noticed that some graphene users find the app running in the background. Others have forced the app to keep running. It doesn't seem to be happening on Pixels and Samsungs.. I am still investigating what could be the issue.   

 I don't think I've ever seen amethyst there, I was just wondering if it might fix some issues.