I’m afraid this is not related to the plugin. Looking at the page source, on the pages where the icons are “missing”, I don’t see the CSS rules declaring the “Font awesome” icon set, and the font-face file is also missing from declaration. The plugin does not use this set, neither does anything with it anywhere in the code.
It is very likely the same cache related issue as the one before. However if you think it is possible to replicate on the test environment, then please add temporary back-end and FTP access to that secondary test site and I will check it directly there. I will try to debug the code then line by line to see what conflicts.