SiteGround looked into it further and got back to me with the following, I can confirm everything work fine when I turn off the ACYMailing component.
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I recreated the problem and I believe that it is somehow related to some of the Acymailing extensions. When I tried to save the changes, the PHP process handling my request started executing a large number of SQL queries like this:
UPDATE q74gf_acymailing_subscriber SET filterflags = REPLACE(filterflags, \"f619f\", \"\") WHERE filterflags LIKE \"%f619f%\"
The number of these queries was so large that until they execute, the script reached the apache timeout. I tried to increase it, but the issue persisted.
The changes started saving immediately and without any error after I disabled all Acymailing related extensions through the URL below:
www.emmett-technique-hq.com/administrator/index.php?option=com_installer&view=manage
Please take a look:
nimb.ws/JVWnMl
You should report the problem to your developer for further investigation.
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So can you please direct me would this be a CBSubs AcyMailing issue or an outright Acymailing support issue?