test: add e2e tests for OpenAI SDK client surviving cache eviction

Add two new e2e tests using real AsyncOpenAI clients:
- test_evicted_openai_sdk_client_stays_usable: verifies size-based eviction
  doesn't close the client
- test_ttl_expired_openai_sdk_client_stays_usable: verifies TTL expiry
  eviction doesn't close the client

Both tests sleep after eviction so any create_task()-based close would
have time to run, making the regression detectable.

Also expand the module docstring to explain why the sleep is required.

tests/test_litellm/caching/test_llm_client_cache_e2e.py

@@ -1,5 +1,19 @@
 """e2e tests: httpx clients obtained via get_async_httpx_client must remain
-usable after LLMClientCache evicts their cache entry."""
+usable after LLMClientCache evicts their cache entry.
+
+These tests exist to prevent a recurring production crash:
+  RuntimeError: Cannot send a request, as the client has been closed.
+
+The bug occurs when LLMClientCache._remove_key() eagerly closes evicted
+clients that are still referenced by in-flight requests.  Every test here
+sleeps after eviction to let the event loop drain any background close
+tasks — a plain ``assert not client.is_closed`` without sleeping is NOT
+sufficient to catch the regression (the close task runs asynchronously).
+
+See: https://github.com/BerriAI/litellm/pull/22247
+"""
+
+import asyncio
 
 import pytest
 
@@ -25,6 +39,10 @@ async def test_evicted_client_is_not_closed():
     # This evicts client_a from cache (capacity=1)
     client_b = get_async_httpx_client(llm_provider="provider_b")
 
+    # Sleep to let any background close tasks execute — without this sleep,
+    # a regression that schedules close via create_task() would go undetected.
+    await asyncio.sleep(0.15)
+
     assert not client_a.client.is_closed
     await client_a.client.aclose()
     await client_b.client.aclose()
@@ -43,5 +61,67 @@ async def test_expired_client_is_not_closed():
         cache.expiration_heap = [(0, key) for _, key in cache.expiration_heap]
     cache.evict_cache()
 
+    await asyncio.sleep(0.15)
+
     assert not client.client.is_closed
     await client.client.aclose()
+
+
+@pytest.mark.asyncio
+async def test_evicted_openai_sdk_client_stays_usable():
+    """OpenAI/Azure SDK clients cached in LLMClientCache must remain usable
+    after eviction.  This is the exact production scenario: the proxy caches
+    an AsyncOpenAI client, the TTL expires, a new request evicts the old
+    entry, but a concurrent streaming request is still reading from it.
+
+    Regression guard: if _remove_key ever calls client.close(), the
+    underlying httpx client is closed and this test fails.
+    """
+    from openai import AsyncOpenAI
+
+    cache = litellm.in_memory_llm_clients_cache
+
+    client = AsyncOpenAI(api_key="sk-test", base_url="https://api.openai.com/v1")
+    cache.set_cache("openai-client", client, ttl=600)
+
+    # Evict by inserting a second entry (max_size=1)
+    cache.set_cache("filler", "x", ttl=600)
+
+    # Let the event loop drain any background close tasks
+    await asyncio.sleep(0.15)
+
+    # The SDK client's internal httpx client must still be open
+    assert not client._client.is_closed, (
+        "AsyncOpenAI client was closed on cache eviction — this causes "
+        "'Cannot send a request, as the client has been closed' in production"
+    )
+    await client.close()
+
+
+@pytest.mark.asyncio
+async def test_ttl_expired_openai_sdk_client_stays_usable():
+    """Same as above but triggered via TTL expiry + get_cache (the other
+    eviction path)."""
+    from openai import AsyncOpenAI
+
+    cache = litellm.in_memory_llm_clients_cache
+
+    client = AsyncOpenAI(api_key="sk-test", base_url="https://api.openai.com/v1")
+    cache.set_cache("openai-client", client, ttl=600)
+
+    # Force TTL expiry
+    for key in list(cache.ttl_dict.keys()):
+        cache.ttl_dict[key] = 0
+        cache.expiration_heap = [(0, k) for _, k in cache.expiration_heap]
+
+    # get_cache calls evict_element_if_expired → _remove_key
+    result = cache.get_cache("openai-client")
+    assert result is None  # expired, so returns None
+
+    await asyncio.sleep(0.15)
+
+    assert not client._client.is_closed, (
+        "AsyncOpenAI client was closed on TTL expiry — this causes "
+        "'Cannot send a request, as the client has been closed' in production"
+    )
+    await client.close()