Giang,
I just tested Autotune in your admin area, and it seems that the fields are being re-fetched each time the FB App page is loaded. My only guess is that's because of the Headline and Description which have special UTF-8 characters in them. Our tables, right now, are created using the default character set for your database. Generally, that's UTF-8, but not always. In the v5.2 release, new installs will always be forced to UTF-8.
I'd recommend trying to remove that Headline/Description from your Facebook app and test if that allows Autotune to complete properly. If so, use phpMySql or another tool to update the #__jfbconnect_config table to be UTF-8.
I hope that helps,
Alex