Here is the JS error I receive:
—
[16:21:45.121] POST http://ablebulgaria.org/test/wp-adminadmin-ajax.php [HTTP/1.1 404 Not Found 924ms]

And here is the correct behavior:
[16:23:07.629] POST http://ablebulgaria.org/test/wp-admin/admin-ajax.php [HTTP/1.1 200 OK 937ms]

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Obviously, the WP Project Manager Fronted URLs are not being processed properly by WPML.