More investigation found that other URL works is due to a flaw in LSWS handling redirect.
when 127.0.0.1 is used, it goes to the Apache default vhost at /var/www/html, the URL does not exist, causes a redirect to /404.shtml, when redirect performed, LSWS uses the "Host" header to locate the vhost and found the cached URL. then purged.
For "/" URL, it does not perform the /404.shtml redirect.
To make it work reliably, add the IP to purge_cache_byurl.php parameter, it works for all URL.