Problem with permissions when supervisor used in php docker image

I am building my app using official php docker image.

Case 1: When I run supervisord with root user and subprocess php-fpm with www-data user. I always get this error log:

FPM initialization failed
failed to open error_log (/proc/self/fd/2): Permission denied (13)

My supervisor config:

[unix_http_server]
file=/run/supervisord.sock  ; the path to the socket file

[supervisord]
logfile=/var/log/supervisord.log ; (main log file;default $CWD/supervisord.log)
pidfile=/var/run/supervisord.pid ; supervisord pidfile; default supervisord.pid

nodaemon=true ; start in foreground if true; default false

[rpcinterface:supervisor]
supervisor.rpcinterface_factory = supervisor.rpcinterface:make_main_rpcinterface

[supervisorctl]
serverurl=unix:///run/supervisord.sock ; use a unix:// URL for a unix socket

[include]
files = /etc/supervisor.d/*.conf

My subprocess php-fpm config:

[program:php-fpm]
process_name=%(program_name)s
command=php-fpm
user=www-data
autostart=true
autorestart=true
redirect_stderr=false
stdout_logfile=/dev/stdout
stdout_logfile_maxbytes=0
stderr_logfile=/dev/stderr
stderr_logfile_maxbytes=0

Note that if I change user of subprocess php-fpm from www-data to root, it worked perfectly

Case 2: When I run supervisord with www-data user (with a little change in config file in order to run it) and subprocess php-fpm also with www-data user, everything works fine.

My supervisor config:

[unix_http_server]
file=/app/supervisord.sock  ; the path to the socket file

[supervisord]
logfile=/app/supervisord.log ; (main log file;default $CWD/supervisord.log)
pidfile=/app/supervisord.pid ; supervisord pidfile; default supervisord.pid
nodaemon=true ; start in foreground if true; default false
user=www-data                     ; setuid to this UNIX account at startup; recommended if root


[rpcinterface:supervisor]
supervisor.rpcinterface_factory = supervisor.rpcinterface:make_main_rpcinterface

[supervisorctl]
serverurl=unix:///run/supervisord.sock ; use a unix:// URL for a unix socket

[include]
files = /etc/supervisor.d/*.conf

My subprocess php-fpm config:

[program:php-fpm]
process_name=%(program_name)s
command=php-fpm
user=www-data
autostart=true
autorestart=true
redirect_stderr=false
stdout_logfile=/dev/stdout
stdout_logfile_maxbytes=0
stderr_logfile=/dev/stderr
stderr_logfile_maxbytes=0

What is the difference between these 2 cases? Thank you!