Queue cron task never runs for phpBB 3.2.
Here https://github.com/phpbb/phpbb/blob/3.2.x/phpBB/includes/functions_messenger.php#L693 the queue cache file set to the ./cache/queue.php.

In the cron task here https://github.com/phpbb/phpbb/blob/3.2.x/phpBB/phpbb/cron/task/core/queue.php#L66 the path is related from the cache directory, which is ./cache/production/ and not just ./cache/.

Thus, expression in is_runnable always returns false, hence the cron task is never runnable.