The proxy was creating a temporary httpx.AsyncClient within an async context manager.
When returning StreamingResponse for SSE endpoints, the client was being closed before
the streaming generator could use it, causing "client has been closed" errors.
This change introduces a shared httpx.AsyncClient that persists for the application
lifecycle, properly cleaned up during shutdown. This also improves performance by
reusing TCP connections across requests.
Co-Authored-By: Claude Sonnet 4.5 <noreply@anthropic.com>
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