Push API setup with Convex

I currently want to implement the Push API with Convex for my PWA.

For that I setup all the neccessary service worker stuff for this to work on the frontend. In my cases with serwist and this example

After that I want to trigger an HTTP Action if a message was created and call inside this action the web-push library.

Is there anything wrong or missing here in my implementation with Convex?
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