Such an issue may appear due to some security plugin, custom function, or rules in your .htaccess file that is preventing the WordPress REST API from working properly. Alternatively, the WordPress cron that manages the compression process is timing out. In order to perform a proper investigation we will need your site URL.
If you need fast assistance, I would recommend creating a ticket from your SiteGround account where my colleagues from the technical support team will assist you as soon as possible.
Thank you I’ve reached out to technical support!