Hi,
Thank you very much for the details.
This was one of the strangest issues I have seen in the last couple of years. So, I used the FTP access to debug through the plugin code, and strangely, the issue boiled down to two WordPress core requests, namely: get_site_option() and update_site_option() calls. The plugin used these functions to save some of the information, and whenever the code reached these, it got stuck for no particular reason. I have replaced these faulty parts with a different solution, now it seems to be okay. I am not sure what exactly the cause is, it looks like something is not okay with those WordPress functions, I might report this to their development team.
Anyways, I will make sure to include this change within the upcoming release, so you won’t have to worry about it anymore. The performance should be all right now.