Supervisord and stdout/stderr
Quoting Supervisord configuration file documentation :
redirect_stderr
If true, cause the process’ stderr output to be sent back to supervisord on its stdout file descriptor (in UNIX shell terms, this is the equivalent of executing/the/program 2>&1).
stderr_logfile
Put process stderr output in this file unless redirect_stderr is true.
Then yes, both stderr and stdout will appear in stdout.log, and stderr.log will remain empty.