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Hello @k0stell0,
Most probably some of the other plugins installed on your site are blocking the REST API, hence the issue. You can check that by examining the browser console for any errors after starting the WebP conversion process.
Additionally, you can check if the cron event that starts the conversion process is presented:
siteground_optimizer_start_webp_conversion_cron
If not, you can restart it using the following command:
wp cron event run siteground_optimizer_start_webp_conversion_cron >/dev/null 2>&1
In case you are still experiencing the issue, please provide us with the URL in question so we can check the case further.