Maybe we can test this idea:

start_bg_task()
yield
SIGTERM_RECEIVED.set()  # Uvicorn will handle SIGTERM on it's own? https://github.com/fastapi/fastapi/issues/2928

Idea behind this:

1. Uvicorn catches SIGTERM
2. Uvicorn triggers lifespan shutdown
3. Background thread finishes current task, sees the event, breaks out
4. Process exits cleanly

This will also be a benefit for usual docker installations

Yep, from my testing, I think we need to wait for the processing to be done in the lifetime handler to avoid it shutting down prematurely.