FastMCP server death on client HEAD calls

Initial Checks

• I confirm that I'm using the latest version of MCP Python SDK
• I confirm that I searched for my issue in https://github.com/modelcontextprotocol/python-sdk/issues before opening this issue

Description

I am using streamable HTTP for MCP. I see HTTP HEAD calls in the log which killing the server. I expect the server to be resilient to poor user behavior.

My code:
https://github.com/deephaven/deephaven-mcp/tree/main/src/deephaven_mcp/mcp_docs_server

Example logs:

2025-08-13 09:29:26.178 PDT
[2025-08-13 16:29:26,179] INFO: [mcp_docs_server:app_lifespan] MCP docs server ready and yielding context
2025-08-13 09:29:26.179 PDT
INFO: 169.254.169.126:65146 - "HEAD /mcp HTTP/1.1" 405 Method Not Allowed
2025-08-13 09:29:26.179 PDT
[2025-08-13 16:29:26,180] INFO: Terminating session: None
2025-08-13 09:29:26.183 PDT
[2025-08-13 16:29:26,180] ERROR: Error in message router
2025-08-13 09:29:26.183 PDT
Traceback (most recent call last):
  File "/app/.venv/lib/python3.12/site-packages/mcp/server/streamable_http.py", line 831, in message_router
    async for session_message in write_stream_reader:
  File "/app/.venv/lib/python3.12/site-packages/anyio/abc/_streams.py", line 41, in __anext__
    return await self.receive()

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^^^^^^^^^^^^^^^^^^^^
2025-08-13 09:29:26.183 PDT
File "/app/.venv/lib/python3.12/site-packages/anyio/streams/memory.py", line 111, in receive
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return self.receive_nowait()
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^^^^^^^^^^^^^^^^^^^^^
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File "/app/.venv/lib/python3.12/site-packages/anyio/streams/memory.py", line 93, in receive_nowait
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raise ClosedResourceError
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anyio.ClosedResourceError
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[2025-08-13 16:29:26,184] INFO: [mcp_docs_server:app_lifespan] Context manager exiting normally
2025-08-13 09:29:26.184 PDT
[2025-08-13 16:29:26,184] INFO: [mcp_docs_server:app_lifespan] Final memory usage: 109.02 MB
2025-08-13 09:29:26.184 PDT
[2025-08-13 16:29:26,184] INFO: [mcp_docs_server:app_lifespan] Final CPU percent: 0.0%
2025-08-13 09:29:26.184 PDT
[2025-08-13 16:29:26,184] INFO: [mcp_docs_server:app_lifespan] Final open file descriptors: 11
2025-08-13 09:29:26.184 PDT
[2025-08-13 16:29:26,185] INFO: [mcp_docs_server:app_lifespan] shutdown active asyncio tasks: 3
2025-08-13 09:29:26.184 PDT
[2025-08-13 16:29:26,185] WARNING: [mcp_docs_server:app_lifespan] Running tasks during shutdown: 3
2025-08-13 09:29:26.184 PDT
[2025-08-13 16:29:26,185] WARNING: [mcp_docs_server:app_lifespan] Running task 1: mcp.server.streamable_http_manager.StreamableHTTPSessionManager._handle_stateless_request.<locals>.run_stateless_server
2025-08-13 09:29:26.184 PDT
[2025-08-13 16:29:26,185] WARNING: [mcp_docs_server:app_lifespan] Running task 2: Task-2
2025-08-13 09:29:26.184 PDT
[2025-08-13 16:29:26,185] WARNING: [mcp_docs_server:app_lifespan] Running task 3: mcp.server.fastmcp.server.FastMCP.run_streamable_http_async
2025-08-13 09:29:26.184 PDT
[2025-08-13 16:29:26,185] INFO: [mcp_docs_server:app_lifespan] shutdown active threads: 1
2025-08-13 09:29:26.185 PDT
[2025-08-13 16:29:26,185] INFO: [mcp_docs_server:app_lifespan] MCP docs server shutting down

My attempts to work around this problem are in this PR. You would need to look at the whole history of what I tried. I included different middleware changes, increasing versions, etc.
https://github.com/deephaven/deephaven-mcp/pull/89

Example Code

My code:
https://github.com/deephaven/deephaven-mcp/tree/main/src/deephaven_mcp/mcp_docs_server

Python & MCP Python SDK

The build can be found at https://github.com/deephaven/deephaven-mcp/blob/main/ops/docker/mcp-docs/Dockerfile.  It is python 3.12 with the latest mcp (version >= 1.12.4).

阅读 mcp/server/streamable_http.py，尤其是 message_router，并跟踪 traceback 中描述的 FastMCP.run_streamable_http_async 入口点。针对 streamable HTTP endpoint 重现一个 HEAD 请求，检查 deephaven-mcp PR #89 中尝试过的 middleware 和版本变更，并验证 server 仍在运行，且不会因意外的 ClosedResourceError shutdown 而停止。