CRIT Supervisor is running as root

Running the cron example leads to the following log message:

CRIT Supervisor is running as root. Privileges were not dropped because no user is specified in the config file. If you intend to run as root, you can set user=root in the config file to avoid this message.

When I change the config to include user=www-data I get an error due to /var/log/supervisord/supervisord.log not being writable (owned by root)

Question is: does supervisord need to run as root? Is is possible to change this without breaking things?
It looks to me like it only executes cron as www-data so it should not require more permissions.

Start with the cron example and its supervisord configuration, then inspect how supervisord runs and where /var/log/supervisord/supervisord.log is created. Determine whether the container requires supervisord to run as root and document or implement a safe configuration that avoids the reported permission error. Done means the cron example runs without the root warning or an unwritable log failure.