Hi Christoffer,
I took the liberty to test the comments in the same page on your live site and found that comments are working fine(refer screenshot). My suspicion is that your test site is calling the wrong url(your live site url) during the ajax call which leads to the 401 Unauthorized error(screenshot2). You should contact your web hosting provider and consult them regarding this issue and see if they can do anything to fix it.
P.S: I am unable to access the backend of your live site so it would be great if you can help me delete those test comments. I apologize for any inconvenience caused.