Cookies werden nicht subdomain-übergreifend geteilt (FastAPI + Nginx)

response.set_cookie(
    key="session_id",
    value=session_id,
    domain=".(meine_domain).me",
    path="/",
    secure=True,
    httponly=True,
    samesite="Lax"
)

location / {
    proxy_pass http://127.0.0.1:8000;
    proxy_set_header Host $host;
    proxy_set_header X-Real-IP $remote_addr;
    proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
    proxy_set_header X-Forwarded-Proto $scheme;
}