Potential race condition in (*mcp.ClientSession).Close()

We've experienced an increasing number of test flakes in the tests for our MCP server. There is an error when closing the connection at the end of the tests: "server is closing: EOF". We think there might be a bug in mcp causing a race between the active part of Close (which breaks the connection and terminates the child process) and the waiting part of close (which seems to observe an error from the terminated child without realizing that close itself was responsible). Alan began tracing the issue here: https://github.com/golang/go/issues/77336#issuecomment-4090756198

To Reproduce
The test is here: https://github.com/golang/tools/blob/36360cdf844bb2553617bdc496c8d972c7229e7d/gopls/internal/cmd/mcp_test.go#L109
Server code is here: https://github.com/golang/tools/blob/master/gopls/internal/cmd/mcp.go
Try to reproduce the failure locally by using -count=100:

go test -run ^TestMCPCommandLogging$ -count=100 golang.org/x/tools/gopls/internal/cmd

=== RUN   TestMCPCommandLogging/trace.log
    gopls: server is closing: EOF
    mcp_test.go:156: closing MCP connection: exit status 2

Expected behavior
Client connection closes without error.

Start with gopls/internal/cmd/mcp_test.go around TestMCPCommandLogging and gopls/internal/cmd/mcp.go, then run go test -run ^TestMCPCommandLogging$ -count=100 golang.org/x/tools/gopls/internal/cmd to reproduce the flake. Trace the MCP client close and child-process termination interaction; done means the client connection closes without error across repeated runs.